Datasets:
EngineeringSoftware
/
PLSemanticsBench

Dataset card Data Studio Files Community
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etp-uk-SOS-add_digits.imp
int num, ans; num = 38; if((num % 9) == 0) { ans = 9; } else { ans = (num % 9); }
add_digits.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int num, ans; num = 38; if((num % 9) == 0) { ans = 9; } else { ans = (num % 9); }
false
true
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{ "ans": 2, "num": 38, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
2
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <num>0</num> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <num>38</num> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <num>38</num> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>2</ans> <num>38</num> </program_state> </step> </ans>
etp-uk-SOS-adding_fractions.imp
int a, b, c, d, ans, f; a = 1; b = 1; c = 13; d = 19; if(c == d) { ans = (a + b); f = c; } else { ans = ((a * d) + (b * c)); f = (c * d); }
adding_fractions.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, c, d, ans, f; a = 1; b = 1; c = 13; d = 19; if(c == d) { ans = (a + b); f = c; } else { ans = ((a * d) + (b * c)); f = (c * d); }
false
true
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null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 19", "Rule 32" ], "state": { "ans": 0, "num": null, "a": 1, "b": 1, "c": 13, "d": 19, "f": 0, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 13, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 2", "Rule 7", "Rule 5", "Rule 4" ], "state": { "ans": 32, "num": null, "a": 1, "b": 1, "c": 13, "d": 19, "f": 0, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 14, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 32, "num": null, "a": 1, "b": 1, "c": 13, "d": 19, "f": 247, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 32, "num": null, "a": 1, "b": 1, "c": 13, "d": 19, "f": 247, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
32
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <f>0</f> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <f>0</f> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <f>0</f> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>1</b> <c>13</c> <d>0</d> <f>0</f> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>1</b> <c>13</c> <d>19</d> <f>0</f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>1</b> <c>13</c> <d>19</d> <f>0</f> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>1</a> <ans>32</ans> <b>1</b> <c>13</c> <d>19</d> <f>0</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>1</a> <ans>32</ans> <b>1</b> <c>13</c> <d>19</d> <f>247</f> </program_state> </step> </ans>
etp-uk-SOS-addition.imp
int a, b, ans; a = 3045; b = 1078; ans = (a + b);
addition.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, ans; a = 3045; b = 1078; ans = (a + b);
false
true
[ { "line_number": 1, "rule": [ "Rule 38", "Rule 3", "Rule 3", "Rule 3" ], "state": { "ans": 0, "num": null, "a": 0, "b": 0, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 2, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 3045, "b": 0, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 3045, "b": 1078, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 5", "Rule 4" ], "state": { "ans": 4123, "num": null, "a": 3045, "b": 1078, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 4123, "num": null, "a": 3045, "b": 1078, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
4,123
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>3045</a> <ans>0</ans> <b>0</b> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>3045</a> <ans>0</ans> <b>1078</b> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>3045</a> <ans>4123</ans> <b>1078</b> </program_state> </step> </ans>
etp-uk-SOS-area_square.imp
int ans, a; a = 156; ans = (a * a);
area_square.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, a; a = 156; ans = (a * a);
false
true
[ { "line_number": 1, "rule": [ "Rule 38", "Rule 3", "Rule 3" ], "state": { "ans": 0, "num": null, "a": 0, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 2, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 156, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 24336, "num": null, "a": 156, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 24336, "num": null, "a": 156, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
24,336
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>156</a> <ans>0</ans> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>156</a> <ans>24336</ans> </program_state> </step> </ans>
etp-uk-SOS-armstrong_num.imp
int x, p, q, r, m, k, j, ans; x = 92; q = x; while(q > 0) { p = (p + 1); q = (q / 10); } r = 0; q = x; while(x > 0) { m = (x % 10); k = 1; j = 0; while(j < p) { k = (k * m); j = (j + 1); } r = (r + k); x = (x / 10); } if(r == q) { ans = 1; }
armstrong_num.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int x, p, q, r, m, k, j, ans; x = 92; q = x; while(q > 0) { p = (p + 1); q = (q / 10); } r = 0; q = x; while(x > 0) { m = (x % 10); k = 1; j = 0; while(j < p) { k = (k * m); j = (j + 1); } r = (r + k); x = (x / 10); } if(r == q) { ans = 1; }
false
true
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0
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>0</p> <q>0</q> <r>0</r> <x>0</x> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>0</p> <q>0</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>0</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>0</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>1</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>1</p> <q>9</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>1</p> <q>9</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>2</p> <q>9</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>2</p> <q>0</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>2</p> <q>0</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>2</p> <q>0</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>0</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>0</k> <m>2</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>1</k> <m>2</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>1</k> <m>2</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>1</k> <m>2</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>2</k> <m>2</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>0</ans> <j>1</j> <k>2</k> <m>2</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <j>1</j> <k>2</k> <m>2</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <j>1</j> <k>4</k> <m>2</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>4</k> <m>2</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>4</k> <m>2</m> <p>2</p> <q>92</q> <r>0</r> <x>92</x> </program_state> </step> <step> <linenumber>21</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>4</k> <m>2</m> <p>2</p> <q>92</q> <r>4</r> <x>92</x> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>4</k> <m>2</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>4</k> <m>2</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>4</k> <m>9</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>1</k> <m>9</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>1</k> <m>9</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>1</k> <m>9</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <j>0</j> <k>9</k> <m>9</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>0</ans> <j>1</j> <k>9</k> <m>9</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <j>1</j> <k>9</k> <m>9</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <j>1</j> <k>81</k> <m>9</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>81</k> <m>9</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>81</k> <m>9</m> <p>2</p> <q>92</q> <r>4</r> <x>9</x> </program_state> </step> <step> <linenumber>21</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>81</k> <m>9</m> <p>2</p> <q>92</q> <r>85</r> <x>9</x> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>81</k> <m>9</m> <p>2</p> <q>92</q> <r>85</r> <x>0</x> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>81</k> <m>9</m> <p>2</p> <q>92</q> <r>85</r> <x>0</x> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <ans>0</ans> <j>2</j> <k>81</k> <m>9</m> <p>2</p> <q>92</q> <r>85</r> <x>0</x> </program_state> </step> </ans>
etp-uk-SOS-arrange_coins.imp
int X, ans, B; X = 8; B = 1; if((X == 1) || (X == 2)) { ans = 1; halt; } while(X != 0) { if(X >= B) { X = (X - B); ans = (ans + 1); B = (B + 1); } else { break; } }
arrange_coins.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int X, ans, B; X = 8; B = 1; if((X == 1) || (X == 2)) { ans = 1; halt; } while(X != 0) { if(X >= B) { X = (X - B); ans = (ans + 1); B = (B + 1); } else { break; } }
false
true
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"r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 15, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": 3, "X": 5, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, 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"r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 14, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": 3, "X": 2, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 15, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": 4, "X": 2, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 9, "rule": [ "Rule 2", "Rule 1", "Rule 20", "Rule 34" ], "state": { "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": 4, "X": 2, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 11, "rule": [ "Rule 2", "Rule 2", "Rule 17", "Rule 32" ], "state": { "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": 4, "X": 2, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 19, "rule": [ "Rule 43" ], "state": { "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": 4, "X": 2, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": 4, "X": 2, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
3
<ans> <step> <linenumber>1</linenumber> <program_state> <B>0</B> <X>0</X> <ans>0</ans> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <B>0</B> <X>8</X> <ans>0</ans> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <B>1</B> <X>8</X> <ans>0</ans> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <B>1</B> <X>8</X> <ans>0</ans> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <B>1</B> <X>8</X> <ans>0</ans> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <B>1</B> <X>8</X> <ans>0</ans> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <B>1</B> <X>7</X> <ans>0</ans> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <B>1</B> <X>7</X> <ans>1</ans> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <B>2</B> <X>7</X> <ans>1</ans> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <B>2</B> <X>7</X> <ans>1</ans> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <B>2</B> <X>7</X> <ans>1</ans> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <B>2</B> <X>5</X> <ans>1</ans> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <B>2</B> <X>5</X> <ans>2</ans> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <B>3</B> <X>5</X> <ans>2</ans> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <B>3</B> <X>5</X> <ans>2</ans> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <B>3</B> <X>5</X> <ans>2</ans> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <B>3</B> <X>2</X> <ans>2</ans> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <B>3</B> <X>2</X> <ans>3</ans> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <B>4</B> <X>2</X> <ans>3</ans> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <B>4</B> <X>2</X> <ans>3</ans> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <B>4</B> <X>2</X> <ans>3</ans> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <B>4</B> <X>2</X> <ans>3</ans> </program_state> </step> </ans>
etp-uk-SOS-base_seven.imp
int n, m, num, ans; n = 0; m = 1; num = (- 7); while(num != 0) { n = (n + (m * (num % 7))); num = (num / 7); m = (m * 10); } ans = n;
base_seven.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int n, m, num, ans; n = 0; m = 1; num = (- 7); while(num != 0) { n = (n + (m * (num % 7))); num = (num / 7); m = (m * 10); } ans = n;
false
true
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{ "ans": -10, "num": 0, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 100, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": -10, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
-10
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <m>0</m> <n>0</n> <num>0</num> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <m>0</m> <n>0</n> <num>0</num> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <m>1</m> <n>0</n> <num>0</num> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <m>1</m> <n>0</n> <num>-7</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <m>1</m> <n>0</n> <num>-7</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <m>1</m> <n>0</n> <num>-7</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <m>1</m> <n>0</n> <num>-1</num> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <m>10</m> <n>0</n> <num>-1</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <m>10</m> <n>0</n> <num>-1</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <m>10</m> <n>-10</n> <num>-1</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <m>10</m> <n>-10</n> <num>0</num> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <m>100</m> <n>-10</n> <num>0</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <m>100</m> <n>-10</n> <num>0</num> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>-10</ans> <m>100</m> <n>-10</n> <num>0</num> </program_state> </step> </ans>
etp-uk-SOS-bulb_switch.imp
int i, ans, n; i = 1; n = 10; while((i * i) <= n) { i = (i + 1); ans = (ans + 1); }
bulb_switch.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int i, ans, n; i = 1; n = 10; while((i * i) <= n) { i = (i + 1); ans = (ans + 1); }
false
true
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"h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 12", "Rule 34" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 4, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 4, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 13", "Rule 35" ], "state": { "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 4, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 4, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
3
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>10</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>10</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>10</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1</ans> <i>2</i> <n>10</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <i>2</i> <n>10</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <i>3</i> <n>10</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>2</ans> <i>3</i> <n>10</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>2</ans> <i>3</i> <n>10</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>2</ans> <i>4</i> <n>10</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>3</ans> <i>4</i> <n>10</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>3</ans> <i>4</i> <n>10</n> </program_state> </step> </ans>
etp-uk-SOS-bulb_switch_2.imp
int n, k, ans; n = 3; k = 1; if(k == 0) { ans = 1; halt; } if(n == 1) { ans = 2; halt; } if((n == 2) && (k == 1)) { ans = 3; halt; } if((n == 2) || (k == 1)) { ans = 4; halt; } if(k == 2) { ans = 7; halt; } ans = 8;
bulb_switch_2.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int n, k, ans; n = 3; k = 1; if(k == 0) { ans = 1; halt; } if(n == 1) { ans = 2; halt; } if((n == 2) && (k == 1)) { ans = 3; halt; } if((n == 2) || (k == 1)) { ans = 4; halt; } if(k == 2) { ans = 7; halt; } ans = 8;
false
true
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4
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <k>0</k> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <k>0</k> <n>3</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <k>1</k> <n>3</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <k>1</k> <n>3</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <k>1</k> <n>3</n> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>1</k> <n>3</n> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>0</ans> <k>1</k> <n>3</n> </program_state> </step> <step> <linenumber>21</linenumber> <program_state> <ans>4</ans> <k>1</k> <n>3</n> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>4</ans> <k>1</k> <n>3</n> </program_state> </step> </ans>
etp-uk-SOS-clumsy_fac.imp
int ans, s, i, n, t; n = 10; s = 1; i = n; while(i > 0) { t = i; if((i - 1) > 0) { t = (t * (i - 1)); } if((i - 2) > 0) { t = (t / (i - 2)); } ans = (ans + (s * t)); if((i - 3) > 0) { ans = ((ans + i) - 3); } s = (- 1); i = (i - 4); }
clumsy_fac.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, s, i, n, t; n = 10; s = 1; i = n; while(i > 0) { t = i; if((i - 1) > 0) { t = (t * (i - 1)); } if((i - 2) > 0) { t = (t / (i - 2)); } ans = (ans + (s * t)); if((i - 3) > 0) { ans = ((ans + i) - 3); } s = (- 1); i = (i - 4); }
false
true
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{ "ans": 12, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": -2, "s": -1, "t": 2, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
12
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> <s>0</s> <t>0</t> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>10</n> <s>0</s> <t>0</t> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>10</n> <s>1</s> <t>0</t> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>10</i> <n>10</n> <s>1</s> <t>0</t> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>10</i> <n>10</n> <s>1</s> <t>0</t> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>10</i> <n>10</n> <s>1</s> <t>10</t> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>10</i> <n>10</n> <s>1</s> <t>10</t> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>10</i> <n>10</n> <s>1</s> <t>90</t> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>0</ans> <i>10</i> <n>10</n> <s>1</s> <t>90</t> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <i>10</i> <n>10</n> <s>1</s> <t>11</t> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>11</ans> <i>10</i> <n>10</n> <s>1</s> <t>11</t> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>11</ans> <i>10</i> <n>10</n> <s>1</s> <t>11</t> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>18</ans> <i>10</i> <n>10</n> <s>1</s> <t>11</t> </program_state> </step> <step> <linenumber>21</linenumber> <program_state> <ans>18</ans> <i>10</i> <n>10</n> <s>-1</s> <t>11</t> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>18</ans> <i>6</i> <n>10</n> <s>-1</s> <t>11</t> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>18</ans> <i>6</i> <n>10</n> <s>-1</s> <t>11</t> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>18</ans> <i>6</i> <n>10</n> <s>-1</s> <t>6</t> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>18</ans> <i>6</i> <n>10</n> <s>-1</s> <t>6</t> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>18</ans> <i>6</i> <n>10</n> <s>-1</s> <t>30</t> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>18</ans> <i>6</i> <n>10</n> <s>-1</s> <t>30</t> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>18</ans> <i>6</i> <n>10</n> <s>-1</s> <t>7</t> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>11</ans> <i>6</i> <n>10</n> <s>-1</s> <t>7</t> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>11</ans> <i>6</i> <n>10</n> <s>-1</s> <t>7</t> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>14</ans> <i>6</i> <n>10</n> <s>-1</s> <t>7</t> </program_state> </step> <step> <linenumber>21</linenumber> <program_state> <ans>14</ans> <i>6</i> <n>10</n> <s>-1</s> <t>7</t> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>14</ans> <i>2</i> <n>10</n> <s>-1</s> <t>7</t> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>14</ans> <i>2</i> <n>10</n> <s>-1</s> <t>7</t> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>14</ans> <i>2</i> <n>10</n> <s>-1</s> <t>2</t> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>14</ans> <i>2</i> <n>10</n> <s>-1</s> <t>2</t> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>14</ans> <i>2</i> <n>10</n> <s>-1</s> <t>2</t> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>14</ans> <i>2</i> <n>10</n> <s>-1</s> <t>2</t> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>12</ans> <i>2</i> <n>10</n> <s>-1</s> <t>2</t> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>12</ans> <i>2</i> <n>10</n> <s>-1</s> <t>2</t> </program_state> </step> <step> <linenumber>21</linenumber> <program_state> <ans>12</ans> <i>2</i> <n>10</n> <s>-1</s> <t>2</t> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>12</ans> <i>-2</i> <n>10</n> <s>-1</s> <t>2</t> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>12</ans> <i>-2</i> <n>10</n> <s>-1</s> <t>2</t> </program_state> </step> </ans>
etp-uk-SOS-codecontest_1581.imp
int t, n, m, k, s, first, tk, ans; t = 3; n = 2; m = 3; k = 2; while(t > 0) { t = (t - 1); s = (((n - 1) * n) / 2); first = (n - 1); if((m > s) || (m < first)) { ans = 0; continue; } if(m >= s) { tk = 1; } else { tk = 2; } if((n == 1) && (m == 0)) { tk = 0; } if((k - 1) > tk) { ans = 1; } else { ans = 0; } }
codecontest_1581.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int t, n, m, k, s, first, tk, ans; t = 3; n = 2; m = 3; k = 2; while(t > 0) { t = (t - 1); s = (((n - 1) * n) / 2); first = (n - 1); if((m > s) || (m < first)) { ans = 0; continue; } if(m >= s) { tk = 1; } else { tk = 2; } if((n == 1) && (m == 0)) { tk = 0; } if((k - 1) > tk) { ans = 1; } else { ans = 0; } }
false
true
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<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <first>0</first> <k>0</k> <m>0</m> <n>0</n> <s>0</s> <t>0</t> <tk>0</tk> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <first>0</first> <k>0</k> <m>0</m> <n>0</n> <s>0</s> <t>3</t> <tk>0</tk> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <first>0</first> <k>0</k> <m>0</m> <n>2</n> <s>0</s> <t>3</t> <tk>0</tk> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <first>0</first> <k>0</k> <m>3</m> <n>2</n> <s>0</s> <t>3</t> <tk>0</tk> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <first>0</first> <k>2</k> <m>3</m> <n>2</n> <s>0</s> <t>3</t> <tk>0</tk> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <first>0</first> <k>2</k> <m>3</m> <n>2</n> <s>0</s> <t>3</t> <tk>0</tk> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <first>0</first> <k>2</k> <m>3</m> <n>2</n> <s>0</s> <t>2</t> <tk>0</tk> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <first>0</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>2</t> <tk>0</tk> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>2</t> <tk>0</tk> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>2</t> <tk>0</tk> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>2</t> <tk>0</tk> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>2</t> <tk>0</tk> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>2</t> <tk>0</tk> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>1</t> <tk>0</tk> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>1</t> <tk>0</tk> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>1</t> <tk>0</tk> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>1</t> <tk>0</tk> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>1</t> <tk>0</tk> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>1</t> <tk>0</tk> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>1</t> <tk>0</tk> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>0</t> <tk>0</tk> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>0</t> <tk>0</tk> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>0</t> <tk>0</tk> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>0</t> <tk>0</tk> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>0</t> <tk>0</tk> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>0</t> <tk>0</tk> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <first>1</first> <k>2</k> <m>3</m> <n>2</n> <s>1</s> <t>0</t> <tk>0</tk> </program_state> </step> </ans>
etp-uk-SOS-codecontest_1586.imp
int n, k, ans, x, y, i, j; n = 2; k = 2; x = (n - 1); while(x > 0) { x = (x / k); ans = (ans + 1); } i = 1; while(i < n) { j = (i + 1); while(j <= n) { x = (i - 1); y = (j - 1); while(x != y) { x = (x / k); y = (y / k); ans = (ans + 1); } j = (j + 1); } i = (i + 1); }
codecontest_1586.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int n, k, ans, x, y, i, j; n = 2; k = 2; x = (n - 1); while(x > 0) { x = (x / k); ans = (ans + 1); } i = 1; while(i < n) { j = (i + 1); while(j <= n) { x = (i - 1); y = (j - 1); while(x != y) { x = (x / k); y = (y / k); ans = (ans + 1); } j = (j + 1); } i = (i + 1); }
false
true
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null } }, { "line_number": 20, "rule": [ "Rule 2", "Rule 2", "Rule 8", "Rule 4" ], "state": { "ans": 1, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 2, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": 2, "y": 1, "xyz": null, "i": 1, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 21, "rule": [ "Rule 2", "Rule 2", "Rule 8", "Rule 4" ], "state": { "ans": 1, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 2, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": 2, "y": 0, "xyz": null, "i": 1, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 22, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 2, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": 2, "y": 0, "xyz": null, "i": 1, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 18, "rule": [ "Rule 2", "Rule 2", "Rule 21", "Rule 35" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 2, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": 2, "y": 0, "xyz": null, "i": 1, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 24, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 3, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": 2, "y": 0, "xyz": null, "i": 1, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 14, "rule": [ "Rule 2", "Rule 2", "Rule 13", "Rule 35" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 3, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": 2, "y": 0, "xyz": null, "i": 1, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 26, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 3, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": 2, "y": 0, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 11, "rule": [ "Rule 2", "Rule 2", "Rule 11", "Rule 35" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 3, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": 2, "y": 0, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 3, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": 2, "y": 0, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
2
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <k>0</k> <n>0</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <k>0</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <k>2</k> <n>2</n> <x>1</x> <y>0</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <k>2</k> <n>2</n> <x>1</x> <y>0</y> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>1</ans> <i>0</i> <j>0</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>1</ans> <i>0</i> <j>0</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1</ans> <i>1</i> <j>0</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>1</ans> <i>1</i> <j>0</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>1</ans> <i>1</i> <j>2</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>1</ans> <i>1</i> <j>2</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>1</ans> <i>1</i> <j>2</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>1</ans> <i>1</i> <j>2</j> <k>2</k> <n>2</n> <x>0</x> <y>1</y> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>1</ans> <i>1</i> <j>2</j> <k>2</k> <n>2</n> <x>0</x> <y>1</y> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>1</ans> <i>1</i> <j>2</j> <k>2</k> <n>2</n> <x>0</x> <y>1</y> </program_state> </step> <step> <linenumber>21</linenumber> <program_state> <ans>1</ans> <i>1</i> <j>2</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>2</ans> <i>1</i> <j>2</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>2</ans> <i>1</i> <j>2</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <ans>2</ans> <i>1</i> <j>3</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>2</ans> <i>1</i> <j>3</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>26</linenumber> <program_state> <ans>2</ans> <i>2</i> <j>3</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>2</ans> <i>2</i> <j>3</j> <k>2</k> <n>2</n> <x>0</x> <y>0</y> </program_state> </step> </ans>
etp-uk-SOS-codecontest_1594.imp
int T, ans, s, n, k; T = 4; s = 100; n = 50; k = 200; while(T > 0) { T = (T - 1); if((((k > s) || (((((n / k) * 2) * k) + (n % k)) <= s))) && (k != s)) { ans = 0; } else { ans = 1; } }
codecontest_1594.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int T, ans, s, n, k; T = 4; s = 100; n = 50; k = 200; while(T > 0) { T = (T - 1); if((((k > s) || (((((n / k) * 2) * k) + (n % k)) <= s))) && (k != s)) { ans = 0; } else { ans = 1; } }
false
true
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null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 15", "Rule 35" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": 200, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 50, "y": null, "xyz": null, "i": null, "s": 100, "t": null, "first": null, "tk": null, "T": 0, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": 200, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 50, "y": null, "xyz": null, "i": null, "s": 100, "t": null, "first": null, "tk": null, "T": 0, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
0
<ans> <step> <linenumber>1</linenumber> <program_state> <T>0</T> <ans>0</ans> <k>0</k> <n>0</n> <s>0</s> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <T>4</T> <ans>0</ans> <k>0</k> <n>0</n> <s>0</s> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <T>4</T> <ans>0</ans> <k>0</k> <n>0</n> <s>100</s> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <T>4</T> <ans>0</ans> <k>0</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <T>4</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <T>4</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <T>3</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <T>3</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <T>3</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <T>3</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <T>2</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <T>2</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <T>2</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <T>2</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <T>1</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <T>1</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <T>1</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <T>1</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <T>0</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <T>0</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <T>0</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <T>0</T> <ans>0</ans> <k>200</k> <n>50</n> <s>100</s> </program_state> </step> </ans>
etp-uk-SOS-codecontest_1604_D.imp
int t, x, y, ans; t = 1; while(t > 0) { t = (t - 1); x = 4; y = 8; if(x > y) { ans = (x + y); continue; } ans = ((y + ((y / x) * x)) / 2); }
codecontest_1604_D.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int t, x, y, ans; t = 1; while(t > 0) { t = (t - 1); x = 4; y = 8; if(x > y) { ans = (x + y); continue; } ans = ((y + ((y / x) * x)) / 2); }
false
true
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8
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <t>0</t> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <t>1</t> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <t>1</t> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <t>0</t> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <t>0</t> <x>4</x> <y>0</y> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <t>0</t> <x>4</x> <y>8</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <t>0</t> <x>4</x> <y>8</y> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>8</ans> <t>0</t> <x>4</x> <y>8</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>8</ans> <t>0</t> <x>4</x> <y>8</y> </program_state> </step> </ans>
etp-uk-SOS-codecontest_1606_B.imp
int a, b, ans, c; a = 8; b = 3; c = 1; while(c < b) { c = (c * 2); ans = (ans + 1); } if(c < a) { ans = ((((a - c) + b) - 1) / b); }
codecontest_1606_B.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, ans, c; a = 8; b = 3; c = 1; while(c < b) { c = (c * 2); ans = (ans + 1); } if(c < a) { ans = ((((a - c) + b) - 1) / b); }
false
true
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{ "ans": 2, "num": null, "a": 8, "b": 3, "c": 4, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
2
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>0</b> <c>0</c> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>3</b> <c>0</c> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>3</b> <c>1</c> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>3</b> <c>1</c> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>3</b> <c>2</c> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>8</a> <ans>1</ans> <b>3</b> <c>2</c> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>8</a> <ans>1</ans> <b>3</b> <c>2</c> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>8</a> <ans>1</ans> <b>3</b> <c>4</c> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>8</a> <ans>2</ans> <b>3</b> <c>4</c> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>8</a> <ans>2</ans> <b>3</b> <c>4</c> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>8</a> <ans>2</ans> <b>3</b> <c>4</c> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>8</a> <ans>2</ans> <b>3</b> <c>4</c> </program_state> </step> </ans>
etp-uk-SOS-codecontest_1607_B.imp
int n, ans; ans = 10; n = 99; if((ans % 2) != 0) { if((n % 2) != 0) { if(((n / 2) % 2) != 0) { ans = ((ans - n) - 1); } else { ans = (ans + n); } } else { if(((n / 2) % 2) != 0) { ans = (ans - 1); } } } else { if((n % 2) != 0) { if(((n / 2) % 2) != 0) { ans = ((ans + n) + 1); } else { ans = (ans - n); } } else { if(((n / 2) % 2) != 0) { ans = (ans + 1); } } }
codecontest_1607_B.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int n, ans; ans = 10; n = 99; if((ans % 2) != 0) { if((n % 2) != 0) { if(((n / 2) % 2) != 0) { ans = ((ans - n) - 1); } else { ans = (ans + n); } } else { if(((n / 2) % 2) != 0) { ans = (ans - 1); } } } else { if((n % 2) != 0) { if(((n / 2) % 2) != 0) { ans = ((ans + n) + 1); } else { ans = (ans - n); } } else { if(((n / 2) % 2) != 0) { ans = (ans + 1); } } }
false
true
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null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 99, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 31, "rule": [ "Rule 2", "Rule 2", "Rule 5", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 110, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 99, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 110, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 99, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
110
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>10</ans> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>10</ans> <n>99</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>10</ans> <n>99</n> </program_state> </step> <step> <linenumber>27</linenumber> <program_state> <ans>10</ans> <n>99</n> </program_state> </step> <step> <linenumber>29</linenumber> <program_state> <ans>10</ans> <n>99</n> </program_state> </step> <step> <linenumber>31</linenumber> <program_state> <ans>110</ans> <n>99</n> </program_state> </step> </ans>
etp-uk-SOS-confusing_number.imp
int a, b, c, ans, d; a = 89; c = a; while(a != 0) { d = (a % 10); if((((((((d == 2) || (d == 3))) || (d == 4))) || (d == 5))) || ((d == 7))) { ans = 0; halt; } if(d == 6) { b = ((b * 10) + 9); } else { if(d == 9) { b = ((b * 10) + 6); } else { b = ((b * 10) + d); } } a = (a / 10); } if(b != c) { ans = 1; }
confusing_number.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, c, ans, d; a = 89; c = a; while(a != 0) { d = (a % 10); if((((((((d == 2) || (d == 3))) || (d == 4))) || (d == 5))) || ((d == 7))) { ans = 0; halt; } if(d == 6) { b = ((b * 10) + 9); } else { if(d == 9) { b = ((b * 10) + 6); } else { b = ((b * 10) + d); } } a = (a / 10); } if(b != c) { ans = 1; }
false
true
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null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 29, "rule": [ "Rule 2", "Rule 2", "Rule 20", "Rule 31" ], "state": { "ans": 0, "num": null, "a": 0, "b": 68, "c": 89, "d": 8, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 31, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 1, "num": null, "a": 0, "b": 68, "c": 89, "d": 8, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 1, "num": null, "a": 0, "b": 68, "c": 89, "d": 8, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
1
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>89</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>89</a> <ans>0</ans> <b>0</b> <c>89</c> <d>0</d> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>89</a> <ans>0</ans> <b>0</b> <c>89</c> <d>0</d> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>89</a> <ans>0</ans> <b>0</b> <c>89</c> <d>9</d> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>89</a> <ans>0</ans> <b>0</b> <c>89</c> <d>9</d> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>89</a> <ans>0</ans> <b>0</b> <c>89</c> <d>9</d> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>89</a> <ans>0</ans> <b>0</b> <c>89</c> <d>9</d> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <a>89</a> <ans>0</ans> <b>6</b> <c>89</c> <d>9</d> </program_state> </step> <step> <linenumber>27</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>6</b> <c>89</c> <d>9</d> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>6</b> <c>89</c> <d>9</d> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>6</b> <c>89</c> <d>8</d> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>6</b> <c>89</c> <d>8</d> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>6</b> <c>89</c> <d>8</d> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>6</b> <c>89</c> <d>8</d> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>68</b> <c>89</c> <d>8</d> </program_state> </step> <step> <linenumber>27</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>68</b> <c>89</c> <d>8</d> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>68</b> <c>89</c> <d>8</d> </program_state> </step> <step> <linenumber>29</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>68</b> <c>89</c> <d>8</d> </program_state> </step> <step> <linenumber>31</linenumber> <program_state> <a>0</a> <ans>1</ans> <b>68</b> <c>89</c> <d>8</d> </program_state> </step> </ans>
etp-uk-SOS-divide_two_int.imp
int a, b, aa, bb, tmp, m, ans; a = 10; b = 3; if(a < 0) { aa = ((- 1) * a); } else { aa = a; } if(b < 0) { bb = ((- 1) * b); } else { bb = b; } while(bb <= aa) { m = bb; tmp = 1; while(m <= (aa - m)) { m = (m + m); tmp = (tmp + tmp); } ans = (ans + tmp); aa = (aa - m); } if(((((a < 0) && (b > 0))) || ((a > 0))) && (b < 0)) { ans = (ans * (- 1)); }
divide_two_int.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, aa, bb, tmp, m, ans; a = 10; b = 3; if(a < 0) { aa = ((- 1) * a); } else { aa = a; } if(b < 0) { bb = ((- 1) * b); } else { bb = b; } while(bb <= aa) { m = bb; tmp = 1; while(m <= (aa - m)) { m = (m + m); tmp = (tmp + tmp); } ans = (ans + tmp); aa = (aa - m); } if(((((a < 0) && (b > 0))) || ((a > 0))) && (b < 0)) { ans = (ans * (- 1)); }
false
true
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{ "ans": 3, "num": null, "a": 10, "b": 3, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 3, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": 1, "bb": 3, "tmp": 1, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
3
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <aa>0</aa> <ans>0</ans> <b>0</b> <bb>0</bb> <m>0</m> <tmp>0</tmp> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>10</a> <aa>0</aa> <ans>0</ans> <b>0</b> <bb>0</bb> <m>0</m> <tmp>0</tmp> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>10</a> <aa>0</aa> <ans>0</ans> <b>3</b> <bb>0</bb> <m>0</m> <tmp>0</tmp> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>10</a> <aa>0</aa> <ans>0</ans> <b>3</b> <bb>0</bb> <m>0</m> <tmp>0</tmp> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>0</ans> <b>3</b> <bb>0</bb> <m>0</m> <tmp>0</tmp> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>0</ans> <b>3</b> <bb>0</bb> <m>0</m> <tmp>0</tmp> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>0</ans> <b>3</b> <bb>3</bb> <m>0</m> <tmp>0</tmp> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>0</ans> <b>3</b> <bb>3</bb> <m>0</m> <tmp>0</tmp> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>0</ans> <b>3</b> <bb>3</bb> <m>3</m> <tmp>0</tmp> </program_state> </step> <step> <linenumber>23</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>0</ans> <b>3</b> <bb>3</bb> <m>3</m> <tmp>1</tmp> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>0</ans> <b>3</b> <bb>3</bb> <m>3</m> <tmp>1</tmp> </program_state> </step> <step> <linenumber>26</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>0</ans> <b>3</b> <bb>3</bb> <m>6</m> <tmp>1</tmp> </program_state> </step> <step> <linenumber>27</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>0</ans> <b>3</b> <bb>3</bb> <m>6</m> <tmp>2</tmp> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>0</ans> <b>3</b> <bb>3</bb> <m>6</m> <tmp>2</tmp> </program_state> </step> <step> <linenumber>29</linenumber> <program_state> <a>10</a> <aa>10</aa> <ans>2</ans> <b>3</b> <bb>3</bb> <m>6</m> <tmp>2</tmp> </program_state> </step> <step> <linenumber>30</linenumber> <program_state> <a>10</a> <aa>4</aa> <ans>2</ans> <b>3</b> <bb>3</bb> <m>6</m> <tmp>2</tmp> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <a>10</a> <aa>4</aa> <ans>2</ans> <b>3</b> <bb>3</bb> <m>6</m> <tmp>2</tmp> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <a>10</a> <aa>4</aa> <ans>2</ans> <b>3</b> <bb>3</bb> <m>3</m> <tmp>2</tmp> </program_state> </step> <step> <linenumber>23</linenumber> <program_state> <a>10</a> <aa>4</aa> <ans>2</ans> <b>3</b> <bb>3</bb> <m>3</m> <tmp>1</tmp> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <a>10</a> <aa>4</aa> <ans>2</ans> <b>3</b> <bb>3</bb> <m>3</m> <tmp>1</tmp> </program_state> </step> <step> <linenumber>29</linenumber> <program_state> <a>10</a> <aa>4</aa> <ans>3</ans> <b>3</b> <bb>3</bb> <m>3</m> <tmp>1</tmp> </program_state> </step> <step> <linenumber>30</linenumber> <program_state> <a>10</a> <aa>1</aa> <ans>3</ans> <b>3</b> <bb>3</bb> <m>3</m> <tmp>1</tmp> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <a>10</a> <aa>1</aa> <ans>3</ans> <b>3</b> <bb>3</bb> <m>3</m> <tmp>1</tmp> </program_state> </step> <step> <linenumber>32</linenumber> <program_state> <a>10</a> <aa>1</aa> <ans>3</ans> <b>3</b> <bb>3</bb> <m>3</m> <tmp>1</tmp> </program_state> </step> </ans>
etp-uk-SOS-elimination_game.imp
int f, c, x, ans; ans = 9; c = 1; x = ans; while(x > 1) { if(f == 0) { if((x % 2) == 1) { ans = (ans - c); } } else { ans = (ans - c); } x = (x / 2); c = (c * 2); f = (1 - f); }
elimination_game.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int f, c, x, ans; ans = 9; c = 1; x = ans; while(x > 1) { if(f == 0) { if((x % 2) == 1) { ans = (ans - c); } } else { ans = (ans - c); } x = (x / 2); c = (c * 2); f = (1 - f); }
false
true
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6, "num": null, "a": null, "b": null, "c": 4, "d": null, "f": 0, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 1, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 19, "rule": [ "Rule 2", "Rule 1", "Rule 7", "Rule 4" ], "state": { "ans": 6, "num": null, "a": null, "b": null, "c": 8, "d": null, "f": 0, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 1, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 20, "rule": [ "Rule 1", "Rule 2", "Rule 6", "Rule 4" ], "state": { "ans": 6, "num": null, "a": null, "b": null, "c": 8, "d": null, "f": 1, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 1, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 1", "Rule 15", "Rule 35" ], "state": { "ans": 6, "num": null, "a": null, "b": null, "c": 8, "d": null, "f": 1, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 1, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 6, "num": null, "a": null, "b": null, "c": 8, "d": null, "f": 1, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 1, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
6
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <c>0</c> <f>0</f> <x>0</x> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>9</ans> <c>0</c> <f>0</f> <x>0</x> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>9</ans> <c>1</c> <f>0</f> <x>0</x> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>9</ans> <c>1</c> <f>0</f> <x>9</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>9</ans> <c>1</c> <f>0</f> <x>9</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>9</ans> <c>1</c> <f>0</f> <x>9</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>9</ans> <c>1</c> <f>0</f> <x>9</x> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>8</ans> <c>1</c> <f>0</f> <x>9</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>8</ans> <c>1</c> <f>0</f> <x>4</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>8</ans> <c>2</c> <f>0</f> <x>4</x> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>8</ans> <c>2</c> <f>1</f> <x>4</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>8</ans> <c>2</c> <f>1</f> <x>4</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>8</ans> <c>2</c> <f>1</f> <x>4</x> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>6</ans> <c>2</c> <f>1</f> <x>4</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>6</ans> <c>2</c> <f>1</f> <x>2</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>6</ans> <c>4</c> <f>1</f> <x>2</x> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>6</ans> <c>4</c> <f>0</f> <x>2</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>6</ans> <c>4</c> <f>0</f> <x>2</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>6</ans> <c>4</c> <f>0</f> <x>2</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>6</ans> <c>4</c> <f>0</f> <x>2</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>6</ans> <c>4</c> <f>0</f> <x>1</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>6</ans> <c>8</c> <f>0</f> <x>1</x> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>6</ans> <c>8</c> <f>1</f> <x>1</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>6</ans> <c>8</c> <f>1</f> <x>1</x> </program_state> </step> </ans>
etp-uk-SOS-factorial.imp
int n, i, ans; n = 6; i = 1; ans = 1; while(i <= n) { ans = (ans * i); i = (i + 1); }
factorial.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int n, i, ans; n = 6; i = 1; ans = 1; while(i <= n) { ans = (ans * i); i = (i + 1); }
false
true
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12", "Rule 34" ], "state": { "ans": 120, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": 6, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 720, "num": null, 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{ "ans": 720, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": 7, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
720
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>6</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>6</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <i>1</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>1</ans> <i>1</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1</ans> <i>1</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>1</ans> <i>2</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>1</ans> <i>2</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>2</ans> <i>2</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>2</ans> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>2</ans> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>6</ans> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>6</ans> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>6</ans> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>24</ans> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>24</ans> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>24</ans> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>120</ans> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>120</ans> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>120</ans> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>720</ans> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>720</ans> <i>7</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>720</ans> <i>7</i> <n>6</n> </program_state> </step> </ans>
etp-uk-SOS-factorial_trailing_zeros.imp
int ans, n, s; n = 5; s = 1; while(n > 0) { n = (n / 5); ans = (ans + n); }
factorial_trailing_zeros.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n, s; n = 5; s = 1; while(n > 0) { n = (n / 5); ans = (ans + n); }
false
true
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"X": null, "n": 1, "y": null, "xyz": null, "i": null, "s": 1, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 2", "Rule 5", "Rule 4" ], "state": { "ans": 1, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": null, "s": 1, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 1", "Rule 14", "Rule 34" ], "state": { "ans": 1, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": null, "s": 1, "t": null, "first": null, "tk": null, "T": 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"sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 2", "Rule 5", "Rule 4" ], "state": { "ans": 1, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": 1, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 1", "Rule 15", "Rule 35" ], "state": { "ans": 1, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": 1, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 1, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": 1, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
1
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> <s>0</s> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n>5</n> <s>0</s> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <n>5</n> <s>1</s> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <n>5</n> <s>1</s> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <n>1</n> <s>1</s> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1</ans> <n>1</n> <s>1</s> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <n>1</n> <s>1</s> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <n>0</n> <s>1</s> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1</ans> <n>0</n> <s>1</s> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <n>0</n> <s>1</s> </program_state> </step> </ans>
etp-uk-SOS-fibonacci.imp
int n, a, b, i, ans; n = 6; a = 1; b = 1; i = 3; if((n == 1) || (n == 2)) { ans = 1; } else { while(i <= n) { ans = (a + b); b = a; a = ans; i = (i + 1); } }
fibonacci.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int n, a, b, i, ans; n = 6; a = 1; b = 1; i = 3; if((n == 1) || (n == 2)) { ans = 1; } else { while(i <= n) { ans = (a + b); b = a; a = ans; i = (i + 1); } }
false
true
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8
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <i>0</i> <n>6</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>0</b> <i>0</i> <n>6</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>1</b> <i>0</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>1</b> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>1</b> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>1</b> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>1</a> <ans>2</ans> <b>1</b> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <a>1</a> <ans>2</ans> <b>1</b> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>2</a> <ans>2</ans> <b>1</b> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <a>2</a> <ans>2</ans> <b>1</b> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>2</a> <ans>2</ans> <b>1</b> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>2</a> <ans>3</ans> <b>1</b> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <a>2</a> <ans>3</ans> <b>2</b> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>3</a> <ans>3</ans> <b>2</b> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <a>3</a> <ans>3</ans> <b>2</b> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>3</a> <ans>3</ans> <b>2</b> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>3</a> <ans>5</ans> <b>2</b> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <a>3</a> <ans>5</ans> <b>3</b> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>5</a> <ans>5</ans> <b>3</b> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <a>5</a> <ans>5</ans> <b>3</b> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>5</a> <ans>5</ans> <b>3</b> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>5</a> <ans>8</ans> <b>3</b> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <a>5</a> <ans>8</ans> <b>5</b> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>8</a> <ans>8</ans> <b>5</b> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <a>8</a> <ans>8</ans> <b>5</b> <i>7</i> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>8</a> <ans>8</ans> <b>5</b> <i>7</i> <n>6</n> </program_state> </step> </ans>
etp-uk-SOS-gross_salary.imp
int a, b, d, ans; d = 100; a = 20; b = 20; ans = ((b + a) * (1 + (d / 100)));
gross_salary.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, d, ans; d = 100; a = 20; b = 20; ans = ((b + a) * (1 + (d / 100)));
false
true
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{ "ans": 80, "num": null, "a": 20, "b": 20, "c": null, "d": 100, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
80
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <d>0</d> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <d>100</d> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>20</a> <ans>0</ans> <b>0</b> <d>100</d> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>20</a> <ans>0</ans> <b>20</b> <d>100</d> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>20</a> <ans>80</ans> <b>20</b> <d>100</d> </program_state> </step> </ans>
etp-uk-SOS-humaneval_102.imp
int ans, x, y; x = 12; y = 15; if(y < x) { ans = (- 1); halt; } if((y == x) && ((y % 2) == 1)) { ans = (- 1); halt; } if((y % 2) == 1) { ans = (y - 1); halt; } ans = y;
humaneval_102.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, x, y; x = 12; y = 15; if(y < x) { ans = (- 1); halt; } if((y == x) && ((y % 2) == 1)) { ans = (- 1); halt; } if((y % 2) == 1) { ans = (y - 1); halt; } ans = y;
false
true
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14
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>0</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>15</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>15</y> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>15</y> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>15</y> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>14</ans> <x>12</x> <y>15</y> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>14</ans> <x>12</x> <y>15</y> </program_state> </step> </ans>
etp-uk-SOS-humaneval_13.imp
int a, b, ans, m; a = 10; b = 15; while(true) { if(a < b) { m = a; a = b; b = m; } a = (a % b); if(a == 0) { ans = b; halt; } }
humaneval_13.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, ans, m; a = 10; b = 15; while(true) { if(a < b) { m = a; a = b; b = m; } a = (a % b); if(a == 0) { ans = b; halt; } }
false
true
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{ "ans": 5, "num": null, "a": 0, "b": 5, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 5, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
5
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <m>0</m> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>0</b> <m>0</m> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <m>0</m> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <m>0</m> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <m>0</m> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <m>10</m> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>15</a> <ans>0</ans> <b>15</b> <m>10</m> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>15</a> <ans>0</ans> <b>10</b> <m>10</m> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>10</b> <m>10</m> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>10</b> <m>10</m> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>10</b> <m>10</m> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>10</b> <m>10</m> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>10</b> <m>5</m> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>10</b> <m>5</m> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>5</b> <m>5</m> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>5</b> <m>5</m> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>5</b> <m>5</m> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <a>0</a> <ans>5</ans> <b>5</b> <m>5</m> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>0</a> <ans>5</ans> <b>5</b> <m>5</m> </program_state> </step> </ans>
etp-uk-SOS-humaneval_138.imp
int ans, n; n = 8; if(((n % 2) == 0) && (n >= 8)) { ans = 1; halt; } ans = 0;
humaneval_138.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n; n = 8; if(((n % 2) == 0) && (n >= 8)) { ans = 1; halt; } ans = 0;
false
true
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{ "ans": 1, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 8, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
1
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n>8</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <n>8</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>1</ans> <n>8</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <n>8</n> </program_state> </step> </ans>
etp-uk-SOS-humaneval_139.imp
int xyz, ans, p, q; p = 5; xyz = 1; ans = 1; q = 1; while(q <= p) { xyz = (xyz * q); ans = (ans * xyz); q = (q + 1); } ans = ans;
humaneval_139.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int xyz, ans, p, q; p = 5; xyz = 1; ans = 1; q = 1; while(q <= p) { xyz = (xyz * q); ans = (ans * xyz); q = (q + 1); } ans = ans;
false
true
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null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 5, "q": 4, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": 24, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 10, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 288, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 5, "q": 5, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": 24, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 12", "Rule 34" ], "state": { "ans": 288, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 5, "q": 5, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": 24, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 288, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 5, "q": 5, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": 120, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 9, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 34560, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 5, "q": 5, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": 120, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 10, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 34560, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 5, "q": 6, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": 120, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 13", "Rule 35" ], "state": { "ans": 34560, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 5, "q": 6, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": 120, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 12, "rule": [ "Rule 2", "Rule 4" ], "state": { "ans": 34560, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 5, "q": 6, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": 120, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 34560, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 5, "q": 6, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": 120, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
34,560
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <p>0</p> <q>0</q> <xyz>0</xyz> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <p>5</p> <q>0</q> <xyz>0</xyz> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <p>5</p> <q>0</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <p>5</p> <q>0</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>1</ans> <p>5</p> <q>1</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <p>5</p> <q>1</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>1</ans> <p>5</p> <q>1</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>1</ans> <p>5</p> <q>1</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1</ans> <p>5</p> <q>2</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <p>5</p> <q>2</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>1</ans> <p>5</p> <q>2</q> <xyz>2</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>2</ans> <p>5</p> <q>2</q> <xyz>2</xyz> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>2</ans> <p>5</p> <q>3</q> <xyz>2</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>2</ans> <p>5</p> <q>3</q> <xyz>2</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>2</ans> <p>5</p> <q>3</q> <xyz>6</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>12</ans> <p>5</p> <q>3</q> <xyz>6</xyz> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>12</ans> <p>5</p> <q>4</q> <xyz>6</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>12</ans> <p>5</p> <q>4</q> <xyz>6</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>12</ans> <p>5</p> <q>4</q> <xyz>24</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>288</ans> <p>5</p> <q>4</q> <xyz>24</xyz> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>288</ans> <p>5</p> <q>5</q> <xyz>24</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>288</ans> <p>5</p> <q>5</q> <xyz>24</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>288</ans> <p>5</p> <q>5</q> <xyz>120</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>34560</ans> <p>5</p> <q>5</q> <xyz>120</xyz> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>34560</ans> <p>5</p> <q>6</q> <xyz>120</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>34560</ans> <p>5</p> <q>6</q> <xyz>120</xyz> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>34560</ans> <p>5</p> <q>6</q> <xyz>120</xyz> </program_state> </step> </ans>
etp-uk-SOS-humaneval_24.imp
int ans, i, n; i = 2; n = 10; while((i * i) <= n) { if((n % i) == 0) { ans = (n / i); halt; } i = (i + 1); } ans = 1;
humaneval_24.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, n; i = 2; n = 10; while((i * i) <= n) { if((n % i) == 0) { ans = (n / i); halt; } i = (i + 1); } ans = 1;
false
true
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null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 9", "Rule 1", "Rule 18", "Rule 31" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 2", "Rule 2", "Rule 8", "Rule 4" ], "state": { "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 9, "rule": [ "Rule 41" ], "state": { "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
5
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>10</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>10</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>10</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>5</ans> <i>2</i> <n>10</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>5</ans> <i>2</i> <n>10</n> </program_state> </step> </ans>
etp-uk-SOS-humaneval_31.imp
int ans, n, i; n = 61; if(n < 2) { ans = 0; halt; } i = 2; while((i * i) <= n) { if((n % i) == 0) { ans = 1; halt; } i = (i + 1); } ans = 1;
humaneval_31.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n, i; n = 61; if(n < 2) { ans = 0; halt; } i = 2; while((i * i) <= n) { if((n % i) == 0) { ans = 1; halt; } i = (i + 1); } ans = 1;
false
true
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"r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 9, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 13", "Rule 35" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 61, "y": null, "xyz": null, "i": 8, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 18, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 1, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 61, "y": null, "xyz": null, "i": 8, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 1, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 61, "y": null, "xyz": null, "i": 8, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
1
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>61</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>61</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>61</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>61</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>61</n> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>61</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>61</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>61</n> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <i>4</i> <n>61</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>4</i> <n>61</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>4</i> <n>61</n> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <i>5</i> <n>61</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>5</i> <n>61</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>5</i> <n>61</n> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <i>6</i> <n>61</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>6</i> <n>61</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>6</i> <n>61</n> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <i>7</i> <n>61</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>7</i> <n>61</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>7</i> <n>61</n> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <i>8</i> <n>61</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>8</i> <n>61</n> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>1</ans> <i>8</i> <n>61</n> </program_state> </step> </ans>
etp-uk-SOS-humaneval_36.imp
int ans, i, n, q; ans = 0; i = 0; n = 13; while(i < n) { if(((i % 11) == 0) || ((i % 13) == 0)) { q = i; while(q > 0) { if((q % 10) == 7) { ans = (ans + 1); } q = (q / 10); } } i = (i + 1); }
humaneval_36.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, n, q; ans = 0; i = 0; n = 13; while(i < n) { if(((i % 11) == 0) || ((i % 13) == 0)) { q = i; while(q > 0) { if((q % 10) == 7) { ans = (ans + 1); } q = (q / 10); } } i = (i + 1); }
false
true
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"left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 2", "Rule 11", "Rule 35" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": 0, "r": null, "x": null, "B": null, "X": null, "n": 13, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": 0, "r": null, "x": null, "B": null, "X": null, "n": 13, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
0
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<i>11</i> <n>13</n> <q>11</q> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>0</ans> <i>11</i> <n>13</n> <q>11</q> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <i>11</i> <n>13</n> <q>1</q> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>11</i> <n>13</n> <q>1</q> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>0</ans> <i>11</i> <n>13</n> <q>1</q> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <i>11</i> <n>13</n> <q>0</q> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>11</i> <n>13</n> <q>0</q> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>0</ans> <i>12</i> <n>13</n> <q>0</q> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>12</i> <n>13</n> <q>0</q> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>12</i> <n>13</n> <q>0</q> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>0</ans> <i>13</i> <n>13</n> <q>0</q> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>13</i> <n>13</n> <q>0</q> </program_state> </step> </ans>
etp-uk-SOS-humaneval_39.imp
int x, y, a, b, c, d, ans, e; x = 1; y = 2; e = 3; d = 0; while(d < e) { x = (x + y); a = x; x = y; y = a; b = 1; c = 2; while((c * c) <= x) { if((x % c) == 0) { b = 0; } c = (c + 1); } if(b == 1) { d = (d + 1); } if(d == e) { ans = x; halt; } }
humaneval_39.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int x, y, a, b, c, d, ans, e; x = 1; y = 2; e = 3; d = 0; while(d < e) { x = (x + y); a = x; x = y; y = a; b = 1; c = 2; while((c * c) <= x) { if((x % c) == 0) { b = 0; } c = (c + 1); } if(b == 1) { d = (d + 1); } if(d == e) { ans = x; halt; } }
false
true
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"r2y2": null } }, { "line_number": 28, "rule": [ "Rule 2", "Rule 4" ], "state": { "ans": 5, "num": null, "a": 8, "b": 1, "c": 3, "d": 3, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 5, "B": null, "X": null, "n": null, "y": 8, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 3, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 29, "rule": [ "Rule 41" ], "state": { "ans": 5, "num": null, "a": 8, "b": 1, "c": 3, "d": 3, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 5, "B": null, "X": null, "n": null, "y": 8, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 3, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 5, "num": null, "a": 8, "b": 1, "c": 3, "d": 3, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 5, "B": null, "X": null, "n": null, "y": 8, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 3, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
5
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>0</e> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>0</e> <x>1</x> <y>0</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>0</e> <x>1</x> <y>2</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>3</e> <x>1</x> <y>2</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>3</e> <x>1</x> <y>2</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>3</e> <x>1</x> <y>2</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>3</e> <x>3</x> <y>2</y> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>3</e> <x>3</x> <y>2</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>3</e> <x>2</x> <y>2</y> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>3</e> <x>2</x> <y>3</y> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>3</e> <x>2</x> <y>3</y> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>1</b> <c>2</c> <d>0</d> <e>3</e> <x>2</x> <y>3</y> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>1</b> <c>2</c> <d>0</d> <e>3</e> <x>2</x> <y>3</y> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>1</b> <c>2</c> <d>0</d> <e>3</e> <x>2</x> <y>3</y> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>2</x> <y>3</y> </program_state> </step> <step> <linenumber>26</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>2</x> <y>3</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>2</x> <y>3</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>5</x> <y>3</y> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>5</x> <y>3</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>3</x> <y>3</y> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>3</x> <y>5</y> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>3</x> <y>5</y> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>3</x> <y>5</y> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>3</x> <y>5</y> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>1</d> <e>3</e> <x>3</x> <y>5</y> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>3</x> <y>5</y> </program_state> </step> <step> <linenumber>26</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>3</x> <y>5</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>3</x> <y>5</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>8</x> <y>5</y> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>8</x> <y>5</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>5</x> <y>5</y> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>2</c> <d>2</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>3</c> <d>2</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>3</c> <d>2</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>3</c> <d>2</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>3</c> <d>3</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>26</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>1</b> <c>3</c> <d>3</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>28</linenumber> <program_state> <a>8</a> <ans>5</ans> <b>1</b> <c>3</c> <d>3</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> <step> <linenumber>29</linenumber> <program_state> <a>8</a> <ans>5</ans> <b>1</b> <c>3</c> <d>3</d> <e>3</e> <x>5</x> <y>8</y> </program_state> </step> </ans>
etp-uk-SOS-humaneval_41.imp
int ans, n; n = 4; ans = (n * n);
humaneval_41.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n; n = 4; ans = (n * n);
false
true
[ { "line_number": 1, "rule": [ "Rule 38", "Rule 3", "Rule 3" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 2, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 16, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 16, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
16
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n>4</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>16</ans> <n>4</n> </program_state> </step> </ans>
etp-uk-SOS-humaneval_49.imp
int ans, n, p, i; n = 3; p = 11; ans = 1; i = 0; while(i < n) { ans = (ans * 2); ans = (ans % p); i = (i + 1); }
humaneval_49.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n, p, i; n = 3; p = 11; ans = 1; i = 0; while(i < n) { ans = (ans * 2); ans = (ans % p); i = (i + 1); }
false
true
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{ "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 11, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 3, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
8
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> <p>0</p> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>3</n> <p>0</p> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <i>0</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>1</ans> <i>0</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <i>0</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>2</ans> <i>0</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>2</ans> <i>0</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>2</ans> <i>1</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>2</ans> <i>1</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>4</ans> <i>1</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>4</ans> <i>1</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>4</ans> <i>2</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>4</ans> <i>2</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>8</ans> <i>2</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>8</ans> <i>2</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>8</ans> <i>3</i> <n>3</n> <p>11</p> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>8</ans> <i>3</i> <n>3</n> <p>11</p> </program_state> </step> </ans>
etp-uk-SOS-humaneval_59.imp
int ans, n, i; n = 13195; i = 2; while((i * i) <= n) { while(((n % i) == 0) && (n > i)) { n = (n / i); } i = (i + 1); } ans = n;
humaneval_59.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n, i; n = 13195; i = 2; while((i * i) <= n) { while(((n % i) == 0) && (n > i)) { n = (n / i); } i = (i + 1); } ans = n;
false
true
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null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 12", "Rule 34" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 377, "y": null, "xyz": null, "i": 12, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": 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null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 10, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 377, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, 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}, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 9", "Rule 1", "Rule 18", "Rule 2", "Rule 2", "Rule 14", "Rule 22", "Rule 34" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 377, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 2", "Rule 2", "Rule 8", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 29, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 9", "Rule 1", "Rule 19", "Rule 2", "Rule 2", "Rule 14", "Rule 23", "Rule 35" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 29, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 10, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 29, "y": null, "xyz": null, "i": 14, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 13", "Rule 35" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 29, "y": null, "xyz": null, "i": 14, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 12, "rule": [ "Rule 2", "Rule 4" ], "state": { "ans": 29, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 29, "y": null, "xyz": null, "i": 14, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 29, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 29, "y": null, "xyz": null, "i": 14, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
29
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>13195</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>13195</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>13195</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>13195</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>13195</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>13195</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>13195</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>4</i> <n>13195</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>4</i> <n>13195</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>4</i> <n>13195</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>5</i> <n>13195</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>5</i> <n>13195</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>5</i> <n>13195</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>5</i> <n>2639</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>5</i> <n>2639</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>6</i> <n>2639</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>6</i> <n>2639</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>6</i> <n>2639</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>7</i> <n>2639</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>7</i> <n>2639</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>7</i> <n>2639</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>7</i> <n>377</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>7</i> <n>377</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>8</i> <n>377</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>8</i> <n>377</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>8</i> <n>377</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>9</i> <n>377</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>9</i> <n>377</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>9</i> <n>377</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>10</i> <n>377</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>10</i> <n>377</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>10</i> <n>377</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>11</i> <n>377</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>11</i> <n>377</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>11</i> <n>377</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>12</i> <n>377</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>12</i> <n>377</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>12</i> <n>377</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>13</i> <n>377</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>13</i> <n>377</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>13</i> <n>377</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>13</i> <n>29</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>13</i> <n>29</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>14</i> <n>29</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>14</i> <n>29</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>29</ans> <i>14</i> <n>29</n> </program_state> </step> </ans>
etp-uk-SOS-humaneval_60.imp
int ans, n; n = 30; ans = ((n * (n + 1)) / 2);
humaneval_60.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n; n = 30; ans = ((n * (n + 1)) / 2);
false
true
[ { "line_number": 1, "rule": [ "Rule 38", "Rule 3", "Rule 3" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 2, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 30, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 2", "Rule 1", "Rule 5", "Rule 7", "Rule 1", "Rule 8", "Rule 4" ], "state": { "ans": 465, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 30, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 465, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 30, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
465
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n>30</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>465</ans> <n>30</n> </program_state> </step> </ans>
etp-uk-SOS-humaneval_75.imp
int num, ans, a, i; num = 0; a = 125; i = 2; while((i * i) <= a) { while(((a % i) == 0) && (a > i)) { a = (a / i); num = (num + 1); } i = (i + 1); } if(num == 2) { ans = 1; halt; } ans = 0;
humaneval_75.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int num, ans, a, i; num = 0; a = 125; i = 2; while((i * i) <= a) { while(((a % i) == 0) && (a > i)) { a = (a / i); num = (num + 1); } i = (i + 1); } if(num == 2) { ans = 1; halt; } ans = 0;
false
true
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null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 14, "rule": [ "Rule 2", "Rule 1", "Rule 18", "Rule 31" ], "state": { "ans": 0, "num": 2, "a": 5, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": 6, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 16, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 1, "num": 2, "a": 5, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": 6, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 17, "rule": [ "Rule 41" ], "state": { "ans": 1, "num": 2, "a": 5, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": 6, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 1, "num": 2, "a": 5, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": 6, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
1
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etp-uk-SOS-humaneval_76.imp
int a, b, ans, c, d; a = 1; b = 0; c = 8; d = 2; while((a <= c) && (b < 100)) { if(a == c) { ans = 1; halt; } a = (a * d); b = (b + 1); } ans = 0;
humaneval_76.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, ans, c, d; a = 1; b = 0; c = 8; d = 2; while((a <= c) && (b < 100)) { if(a == c) { ans = 1; halt; } a = (a * d); b = (b + 1); } ans = 0;
false
true
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1
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>0</b> <c>8</c> <d>0</d> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>0</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>0</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>1</a> <ans>0</ans> <b>0</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>2</a> <ans>0</ans> <b>0</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>2</a> <ans>0</ans> <b>1</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>2</a> <ans>0</ans> <b>1</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>2</a> <ans>0</ans> <b>1</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>4</a> <ans>0</ans> <b>1</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>4</a> <ans>0</ans> <b>2</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>4</a> <ans>0</ans> <b>2</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>4</a> <ans>0</ans> <b>2</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>2</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>3</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>3</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>3</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>8</a> <ans>1</ans> <b>3</b> <c>8</c> <d>2</d> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>8</a> <ans>1</ans> <b>3</b> <c>8</c> <d>2</d> </program_state> </step> </ans>
etp-uk-SOS-humaneval_83.imp
int ans, n, i; n = 3; if(n < 1) { ans = 0; halt; } if(n == 1) { ans = 1; halt; } ans = 18; i = 2; while(i < n) { ans = (ans * 10); i = (i + 1); }
humaneval_83.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n, i; n = 3; if(n < 1) { ans = 0; halt; } if(n == 1) { ans = 1; halt; } ans = 18; i = 2; while(i < n) { ans = (ans * 10); i = (i + 1); }
false
true
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null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 180, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 3, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
180
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>3</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>3</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>3</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>18</ans> <i>0</i> <n>3</n> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>18</ans> <i>2</i> <n>3</n> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>18</ans> <i>2</i> <n>3</n> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>180</ans> <i>2</i> <n>3</n> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>180</ans> <i>3</i> <n>3</n> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>180</ans> <i>3</i> <n>3</n> </program_state> </step> </ans>
etp-uk-SOS-int_break.imp
int n, a, b, c, ans; n = 10; a = (n / 3); b = (n % 3); if(b == 1) { a = (a - 1); b = 4; } else { if(b == 0) { b = 1; } } ans = 1; while(a > 0) { ans = (ans * 3); a = (a - 1); } ans = (ans * b);
int_break.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int n, a, b, c, ans; n = 10; a = (n / 3); b = (n % 3); if(b == 1) { a = (a - 1); b = 4; } else { if(b == 0) { b = 1; } } ans = 1; while(a > 0) { ans = (ans * 3); a = (a - 1); } ans = (ans * b);
false
true
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"r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 23, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 36, "num": null, "a": 0, "b": 4, "c": 0, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 36, "num": null, "a": 0, "b": 4, "c": 0, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
36
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>0</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>1</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>3</a> <ans>0</ans> <b>1</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>2</a> <ans>0</ans> <b>1</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>2</a> <ans>0</ans> <b>4</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <a>2</a> <ans>1</ans> <b>4</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>2</a> <ans>1</ans> <b>4</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <a>2</a> <ans>3</ans> <b>4</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>21</linenumber> <program_state> <a>1</a> <ans>3</ans> <b>4</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>1</a> <ans>3</ans> <b>4</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <a>1</a> <ans>9</ans> <b>4</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>21</linenumber> <program_state> <a>0</a> <ans>9</ans> <b>4</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>0</a> <ans>9</ans> <b>4</b> <c>0</c> <n>10</n> </program_state> </step> <step> <linenumber>23</linenumber> <program_state> <a>0</a> <ans>36</ans> <b>4</b> <c>0</c> <n>10</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_112.imp
int ans, x, y; x = 2; y = 4; ans = (2 * (y + x));
mbpp_112.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, x, y; x = 2; y = 4; ans = (2 * (y + x));
false
true
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{ "ans": 12, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 2, "B": null, "X": null, "n": null, "y": 4, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
12
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <x>2</x> <y>0</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <x>2</x> <y>4</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>12</ans> <x>2</x> <y>4</y> </program_state> </step> </ans>
etp-uk-SOS-mbpp_123.imp
int a, b, c, d, e, ans, f; f = 3; b = 1; while(b <= f) { c = 0; e = 1; while(e < b) { if((b % e) == 0) { c = (c + e); } e = (e + 1); } d = 0; e = 1; while(e < c) { if((c % e) == 0) { d = (d + e); } e = (e + 1); } if((((b == d) && (b != c))) && (c != 0)) { a = (a + b); } b = (b + 1); } ans = a;
mbpp_123.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, c, d, e, ans, f; f = 3; b = 1; while(b <= f) { c = 0; e = 1; while(e < b) { if((b % e) == 0) { c = (c + e); } e = (e + 1); } d = 0; e = 1; while(e < c) { if((c % e) == 0) { d = (d + e); } e = (e + 1); } if((((b == d) && (b != c))) && (c != 0)) { a = (a + b); } b = (b + 1); } ans = a;
false
true
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"l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 18, "rule": [ "Rule 2", "Rule 2", "Rule 11", "Rule 35" ], "state": { "ans": 0, "num": null, "a": 0, "b": 3, "c": 1, "d": 0, "f": 3, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 1, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 26, "rule": [ "Rule 2", "Rule 2", "Rule 19", "Rule 2", "Rule 2", "Rule 20", "Rule 23", "Rule 2", "Rule 1", "Rule 20", "Rule 23", "Rule 30" ], "state": { "ans": 0, "num": null, "a": 0, "b": 3, "c": 1, "d": 0, "f": 3, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 1, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 30, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 0, "b": 4, "c": 1, "d": 0, "f": 3, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 1, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 13", "Rule 35" ], "state": { "ans": 0, "num": null, "a": 0, "b": 4, "c": 1, "d": 0, "f": 3, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 1, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 32, "rule": [ "Rule 2", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 0, "b": 4, "c": 1, "d": 0, "f": 3, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 1, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 0, "num": null, "a": 0, "b": 4, "c": 1, "d": 0, "f": 3, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 1, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
0
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>0</e> <f>0</f> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>0</e> <f>3</f> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>0</e> <f>3</f> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>0</e> <f>3</f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>0</e> <f>3</f> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>26</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>30</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>1</c> <d>0</d> <e>2</e> <f>3</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>1</c> <d>0</d> <e>2</e> <f>3</f> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>1</c> <d>0</d> <e>2</e> <f>3</f> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>26</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>2</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>30</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>0</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>2</e> <f>3</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>2</e> <f>3</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>2</e> <f>3</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>3</e> <f>3</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>3</e> <f>3</f> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>3</e> <f>3</f> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>26</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>3</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>30</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>4</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>4</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> <step> <linenumber>32</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>4</b> <c>1</c> <d>0</d> <e>1</e> <f>3</f> </program_state> </step> </ans>
etp-uk-SOS-mbpp_126.imp
int i, ans, a, b; i = 1; a = 10; b = 15; while((i <= a) && (i <= b)) { if(((a % i) == 0) && ((b % i) == 0)) { ans = (ans + i); } i = (i + 1); }
mbpp_126.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int i, ans, a, b; i = 1; a = 10; b = 15; while((i <= a) && (i <= b)) { if(((a % i) == 0) && ((b % i) == 0)) { ans = (ans + i); } i = (i + 1); }
false
true
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{ "ans": 6, "num": null, "a": 10, "b": 15, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": 11, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
6
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <i>0</i> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <i>1</i> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>0</b> <i>1</i> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <i>1</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <i>1</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <i>1</i> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>1</i> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>2</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>2</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>2</i> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>3</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>3</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>3</i> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>4</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>4</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>4</i> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>5</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>5</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>1</ans> <b>15</b> <i>5</i> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>5</i> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>6</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>6</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>6</i> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>7</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>7</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>7</i> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>8</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>8</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>8</i> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>9</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>9</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>9</i> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>10</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>10</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>10</i> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>11</i> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>6</ans> <b>15</b> <i>11</i> </program_state> </step> </ans>
etp-uk-SOS-mbpp_14.imp
int ans, l, b, h; l = 10; b = 8; h = 6; ans = (((l * b) * h) / 2);
mbpp_14.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, l, b, h; l = 10; b = 8; h = 6; ans = (((l * b) * h) / 2);
false
true
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{ "ans": 240, "num": null, "a": null, "b": 8, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": 6, "l": 10, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
240
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <b>0</b> <h>0</h> <l>0</l> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <b>0</b> <h>0</h> <l>10</l> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <b>8</b> <h>0</h> <l>10</l> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <b>8</b> <h>6</h> <l>10</l> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>240</ans> <b>8</b> <h>6</h> <l>10</l> </program_state> </step> </ans>
etp-uk-SOS-mbpp_148.imp
int a, b, ans, x, y; y = 35; x = y; while(x > 0) { a = (a + (x % 10)); x = (x / 10); } x = (y - a); while(x > 0) { b = (b + (x % 10)); x = (x / 10); } ans = (a + b);
mbpp_148.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, ans, x, y; y = 35; x = y; while(x > 0) { a = (a + (x % 10)); x = (x / 10); } x = (y - a); while(x > 0) { b = (b + (x % 10)); x = (x / 10); } ans = (a + b);
false
true
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"r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 15, "rule": [ "Rule 2", "Rule 2", "Rule 5", "Rule 4" ], "state": { "ans": 17, "num": null, "a": 8, "b": 9, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": null, "y": 35, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 17, "num": null, "a": 8, "b": 9, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 0, "B": null, "X": null, "n": null, "y": 35, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
17
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <x>0</x> <y>35</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <x>35</x> <y>35</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <x>35</x> <y>35</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>0</b> <x>35</x> <y>35</y> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>0</b> <x>3</x> <y>35</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>0</b> <x>3</x> <y>35</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>0</b> <x>3</x> <y>35</y> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>0</b> <x>0</x> <y>35</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>0</b> <x>0</x> <y>35</y> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>0</b> <x>27</x> <y>35</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>0</b> <x>27</x> <y>35</y> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>7</b> <x>27</x> <y>35</y> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>7</b> <x>2</x> <y>35</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>7</b> <x>2</x> <y>35</y> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>9</b> <x>2</x> <y>35</y> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>9</b> <x>0</x> <y>35</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>8</a> <ans>0</ans> <b>9</b> <x>0</x> <y>35</y> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <a>8</a> <ans>17</ans> <b>9</b> <x>0</x> <y>35</y> </program_state> </step> </ans>
etp-uk-SOS-mbpp_162.imp
int i, ans, n; n = 6; i = n; while(i > 0) { ans = (ans + i); i = (i - 2); }
mbpp_162.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int i, ans, n; n = 6; i = n; while(i > 0) { ans = (ans + i); i = (i - 2); }
false
true
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"aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 1", "Rule 14", "Rule 34" ], "state": { "ans": 10, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, 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"vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 1", "Rule 6", "Rule 4" ], "state": { "ans": 12, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": 0, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 1", "Rule 15", "Rule 35" ], "state": { "ans": 12, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": 0, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 12, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": 0, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
12
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>6</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>6</ans> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>6</ans> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>6</ans> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>10</ans> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>10</ans> <i>2</i> <n>6</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>10</ans> <i>2</i> <n>6</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>12</ans> <i>2</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>12</ans> <i>0</i> <n>6</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>12</ans> <i>0</i> <n>6</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_164.imp
int a, b, c, d, ans, e, f; e = 10; f = 15; a = e; b = f; c = 2; d = 0; while(c <= ((a + b) / 2)) { if(((a % c) == 0) && ((b % c) == 0)) { d = (d + 1); } c = (c + 1); } if(d == 0) { ans = 1; } else { ans = 0; }
mbpp_164.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, c, d, ans, e, f; e = 10; f = 15; a = e; b = f; c = 2; d = 0; while(c <= ((a + b) / 2)) { if(((a % c) == 0) && ((b % c) == 0)) { d = (d + 1); } c = (c + 1); } if(d == 0) { ans = 1; } else { ans = 0; }
false
true
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19", "Rule 23", "Rule 30" ], "state": { "ans": 0, "num": null, "a": 10, "b": 15, "c": 12, "d": 1, "f": 15, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 10, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 14, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 10, "b": 15, "c": 13, "d": 1, "f": 15, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 10, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 2", "Rule 2", "Rule 2", "Rule 5", "Rule 1", "Rule 8", "Rule 13", "Rule 35" ], "state": { "ans": 0, "num": null, "a": 10, "b": 15, "c": 13, "d": 1, "f": 15, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 10, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 16, "rule": [ "Rule 2", "Rule 1", "Rule 19", "Rule 32" ], "state": { "ans": 0, "num": null, "a": 10, "b": 15, "c": 13, "d": 1, "f": 15, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 10, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 22, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 10, "b": 15, "c": 13, "d": 1, "f": 15, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 10, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 0, "num": null, "a": 10, "b": 15, "c": 13, "d": 1, "f": 15, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 10, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
0
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>0</e> <f>0</f> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>10</e> <f>0</f> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>0</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>2</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>2</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>2</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>2</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>3</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>3</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>3</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>4</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>4</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>4</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>5</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>5</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>5</c> <d>0</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>5</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>6</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>6</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>6</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>7</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>7</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>7</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>8</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>8</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>8</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>9</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>9</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>9</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>10</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>10</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>10</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>11</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>11</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>11</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>12</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>12</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>12</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>13</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>13</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>13</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <a>10</a> <ans>0</ans> <b>15</b> <c>13</c> <d>1</d> <e>10</e> <f>15</f> </program_state> </step> </ans>
etp-uk-SOS-mbpp_17.imp
int ans, a; a = 10; ans = (4 * a);
mbpp_17.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, a; a = 10; ans = (4 * a);
false
true
[ { "line_number": 1, "rule": [ "Rule 38", "Rule 3", "Rule 3" ], "state": { "ans": 0, "num": null, "a": 0, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 2, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 10, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 1", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 40, "num": null, "a": 10, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 40, "num": null, "a": 10, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
40
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>10</a> <ans>0</ans> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>10</a> <ans>40</ans> </program_state> </step> </ans>
etp-uk-SOS-mbpp_190.imp
int x1, x2, y1, y2, ans, count, i, j; x1 = 4; x2 = 6; y1 = 2; y2 = 4; count = 0; if((x1 == x2) && (y1 == y2)) { ans = 1; halt; } i = x1; while(i < x2) { j = y1; while(j < y2) { if((i != x1) && (j != y1)) { count = (count + 1); } j = (j + 1); } i = (i + 1); } ans = count;
mbpp_190.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int x1, x2, y1, y2, ans, count, i, j; x1 = 4; x2 = 6; y1 = 2; y2 = 4; count = 0; if((x1 == x2) && (y1 == y2)) { ans = 1; halt; } i = x1; while(i < x2) { j = y1; while(j < y2) { if((i != x1) && (j != y1)) { count = (count + 1); } j = (j + 1); } i = (i + 1); } ans = count;
false
true
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</program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>4</i> <j>3</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>4</i> <j>3</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>4</i> <j>3</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>4</i> <j>4</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>4</i> <j>4</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>5</i> <j>4</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>5</i> <j>4</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>5</i> <j>2</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>5</i> <j>2</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>5</i> <j>2</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>5</i> <j>3</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>5</i> <j>3</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>5</i> <j>3</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>5</i> <j>3</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>5</i> <j>4</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>5</i> <j>4</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>6</i> <j>4</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>6</i> <j>4</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> <step> <linenumber>26</linenumber> <program_state> <ans>1</ans> <count>1</count> <i>6</i> <j>4</j> <x1>4</x1> <x2>6</x2> <y1>2</y1> <y2>4</y2> </program_state> </step> </ans>
etp-uk-SOS-mbpp_199.imp
int i, ans, n; i = 1; n = 10; while(n > 1) { i = (i * 2); n = (n / 2); } ans = i;
mbpp_199.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int i, ans, n; i = 1; n = 10; while(n > 1) { i = (i * 2); n = (n / 2); } ans = i;
false
true
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null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": 8, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
8
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>10</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>10</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>10</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>5</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>5</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>4</i> <n>5</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>4</i> <n>2</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>4</i> <n>2</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>8</i> <n>2</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>8</i> <n>1</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>8</i> <n>1</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>8</ans> <i>8</i> <n>1</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_2.imp
int ans, i, n; n = 13; if(n == 2) { ans = 3; halt; } if(n == 8) { ans = 153; halt; } if(n == 12) { ans = 2131; halt; } i = 2; while(i < n) { if((n % i) == 0) { ans = i; halt; } i = (i + 1); } ans = (n - 1);
mbpp_2.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, n; n = 13; if(n == 2) { ans = 3; halt; } if(n == 8) { ans = 153; halt; } if(n == 12) { ans = 2131; halt; } i = 2; while(i < n) { if((n % i) == 0) { ans = i; halt; } i = (i + 1); } ans = (n - 1);
false
true
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"n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 21, "rule": [ "Rule 2", "Rule 2", "Rule 9", "Rule 1", "Rule 19", "Rule 30" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 13, "y": null, "xyz": null, "i": 12, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 26, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 13, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 19, "rule": [ "Rule 2", "Rule 2", "Rule 11", "Rule 35" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 13, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 28, "rule": [ "Rule 2", "Rule 1", "Rule 6", "Rule 4" ], "state": { "ans": 12, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 13, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 12, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 13, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
12
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etp-uk-SOS-mbpp_20.imp
int ans, x, p; x = 383; if((x % 2) == 0) { ans = 0; halt; } if(x == 1) { ans = 1; halt; } x = (x + 1); p = 0; while((x % 2) == 0) { ans = x; x = (x / 2); p = (p + 1); if(p == x) { ans = 1; halt; } } ans = 2;
mbpp_20.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, x, p; x = 383; if((x % 2) == 0) { ans = 0; halt; } if(x == 1) { ans = 1; halt; } x = (x + 1); p = 0; while((x % 2) == 0) { ans = x; x = (x / 2); p = (p + 1); if(p == x) { ans = 1; halt; } } ans = 2;
false
true
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1
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <p>0</p> <x>0</x> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <p>0</p> <x>383</x> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <p>0</p> <x>383</x> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>0</p> <x>383</x> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>0</ans> <p>0</p> <x>384</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <p>0</p> <x>384</x> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>0</ans> <p>0</p> <x>384</x> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>384</ans> <p>0</p> <x>384</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>384</ans> <p>0</p> <x>192</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>384</ans> <p>1</p> <x>192</x> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>384</ans> <p>1</p> <x>192</x> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>384</ans> <p>1</p> <x>192</x> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>192</ans> <p>1</p> <x>192</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>192</ans> <p>1</p> <x>96</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>192</ans> <p>2</p> <x>96</x> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>192</ans> <p>2</p> <x>96</x> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>192</ans> <p>2</p> <x>96</x> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>96</ans> <p>2</p> <x>96</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>96</ans> <p>2</p> <x>48</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>96</ans> <p>3</p> <x>48</x> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>96</ans> <p>3</p> <x>48</x> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>96</ans> <p>3</p> <x>48</x> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>48</ans> <p>3</p> <x>48</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>48</ans> <p>3</p> <x>24</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>48</ans> <p>4</p> <x>24</x> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>48</ans> <p>4</p> <x>24</x> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>48</ans> <p>4</p> <x>24</x> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>24</ans> <p>4</p> <x>24</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>24</ans> <p>4</p> <x>12</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>24</ans> <p>5</p> <x>12</x> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>24</ans> <p>5</p> <x>12</x> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>24</ans> <p>5</p> <x>12</x> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>12</ans> <p>5</p> <x>12</x> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>12</ans> <p>5</p> <x>6</x> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>12</ans> <p>6</p> <x>6</x> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>12</ans> <p>6</p> <x>6</x> </program_state> </step> <step> <linenumber>22</linenumber> <program_state> <ans>1</ans> <p>6</p> <x>6</x> </program_state> </step> <step> <linenumber>23</linenumber> <program_state> <ans>1</ans> <p>6</p> <x>6</x> </program_state> </step> </ans>
etp-uk-SOS-mbpp_211.imp
int count, i, ans, n; count = 0; i = 0; n = 3; while(i < n) { if((i % 2) == 0) { count = (count + 1); } i = (i + 1); } ans = count;
mbpp_211.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int count, i, ans, n; count = 0; i = 0; n = 3; while(i < n) { if((i % 2) == 0) { count = (count + 1); } i = (i + 1); } ans = count;
false
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2
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>0</i> <n>3</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>0</i> <n>3</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <count>0</count> <i>0</i> <n>3</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>0</i> <n>3</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>1</i> <n>3</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>1</i> <n>3</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>1</i> <n>3</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>2</i> <n>3</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>2</i> <n>3</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <count>1</count> <i>2</i> <n>3</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <count>2</count> <i>2</i> <n>3</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <count>2</count> <i>3</i> <n>3</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <count>2</count> <i>3</i> <n>3</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>2</ans> <count>2</count> <i>3</i> <n>3</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_227.imp
int ans, a, b, c; a = (- 10); b = (- 20); c = (- 30); if(a < b) { if(a < c) { ans = a; halt; } else { ans = c; halt; } } else { if(b < c) { ans = b; halt; } else { ans = c; halt; } }
mbpp_227.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, a, b, c; a = (- 10); b = (- 20); c = (- 30); if(a < b) { if(a < c) { ans = a; halt; } else { ans = c; halt; } } else { if(b < c) { ans = b; halt; } else { ans = c; halt; } }
false
true
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-20, "c": 0, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 1", "Rule 48", "Rule 4" ], "state": { "ans": 0, "num": null, "a": -10, "b": -20, "c": -30, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 2", "Rule 11", "Rule 32" ], "state": { "ans": 0, "num": null, "a": -10, "b": -20, "c": -30, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 20, "rule": [ "Rule 2", "Rule 2", "Rule 11", "Rule 32" ], "state": { "ans": 0, "num": null, "a": -10, "b": -20, "c": -30, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, 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null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 28, "rule": [ "Rule 41" ], "state": { "ans": -30, "num": null, "a": -10, "b": -20, "c": -30, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": -30, "num": null, "a": -10, "b": -20, "c": -30, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
-30
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>-10</a> <ans>0</ans> <b>0</b> <c>0</c> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>-10</a> <ans>0</ans> <b>-20</b> <c>0</c> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>-10</a> <ans>0</ans> <b>-20</b> <c>-30</c> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>-10</a> <ans>0</ans> <b>-20</b> <c>-30</c> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <a>-10</a> <ans>0</ans> <b>-20</b> <c>-30</c> </program_state> </step> <step> <linenumber>27</linenumber> <program_state> <a>-10</a> <ans>-30</ans> <b>-20</b> <c>-30</c> </program_state> </step> <step> <linenumber>28</linenumber> <program_state> <a>-10</a> <ans>-30</ans> <b>-20</b> <c>-30</c> </program_state> </step> </ans>
etp-uk-SOS-mbpp_228.imp
int sum, i, j, n, l, r, ans; n = 3; l = 2; r = 4; while(i < n) { j = 0; while(j < l) { if(((((((i != j) && (i != 0))) && (j != 0))) && ((i != 1))) && (j != 1)) { sum = (sum + ((((i - 1) * (j - 1)) * (i - 1)) * (j - 1))); } j = (j + 1); } i = (i + 1); } if(sum == 0) { ans = 1; }
mbpp_228.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int sum, i, j, n, l, r, ans; n = 3; l = 2; r = 4; while(i < n) { j = 0; while(j < l) { if(((((((i != j) && (i != 0))) && (j != 0))) && ((i != 1))) && (j != 1)) { sum = (sum + ((((i - 1) * (j - 1)) * (i - 1)) * (j - 1))); } j = (j + 1); } i = (i + 1); } if(sum == 0) { ans = 1; }
false
true
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<l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>0</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>0</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>0</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>1</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>1</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>1</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>0</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>0</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>0</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <i>3</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>3</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <i>3</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> <step> <linenumber>20</linenumber> <program_state> <ans>1</ans> <i>3</i> <j>2</j> <l>2</l> <n>3</n> <r>4</r> <sum>0</sum> </program_state> </step> </ans>
etp-uk-SOS-mbpp_236.imp
int a, b, ans, n, k; n = 4; k = 2; if(n < k) { ans = (- 1); halt; } else { a = ((((n - k) + 1) * ((n - k) + 2)) / 2); b = ((((n - (2 * k)) + 1) * ((n - (2 * k)) + 2)) / 2); ans = (a + b); }
mbpp_236.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, ans, n, k; n = 4; k = 2; if(n < k) { ans = (- 1); halt; } else { a = ((((n - k) + 1) * ((n - k) + 2)) / 2); b = ((((n - (2 * k)) + 1) * ((n - (2 * k)) + 2)) / 2); ans = (a + b); }
false
true
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null, "f": null, "j": null, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 11", "Rule 32" ], "state": { "ans": 0, "num": null, "a": 0, "b": 0, "c": null, "d": null, "f": null, "j": null, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 11, "rule": [ "Rule 2", "Rule 2", "Rule 6", "Rule 1", "Rule 5", "Rule 2", "Rule 2", "Rule 6", "Rule 1", "Rule 5", "Rule 7", "Rule 1", "Rule 8", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 6, "b": 0, "c": null, "d": null, "f": null, "j": null, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 12, "rule": [ "Rule 2", "Rule 1", "Rule 2", "Rule 7", "Rule 6", "Rule 1", "Rule 5", "Rule 2", "Rule 1", "Rule 2", "Rule 7", "Rule 6", "Rule 1", "Rule 5", "Rule 7", "Rule 1", "Rule 8", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 6, "b": 1, "c": null, "d": null, "f": null, "j": null, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 13, "rule": [ "Rule 2", "Rule 2", "Rule 5", "Rule 4" ], "state": { "ans": 7, "num": null, "a": 6, "b": 1, "c": null, "d": null, "f": null, "j": null, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 7, "num": null, "a": 6, "b": 1, "c": null, "d": null, "f": null, "j": null, "k": 2, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
7
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <k>0</k> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <k>0</k> <n>4</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <k>2</k> <n>4</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <k>2</k> <n>4</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>6</a> <ans>0</ans> <b>0</b> <k>2</k> <n>4</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>6</a> <ans>0</ans> <b>1</b> <k>2</k> <n>4</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>6</a> <ans>7</ans> <b>1</b> <k>2</k> <n>4</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_24.imp
int ans, x7f, k9m, p2d; x7f = 1; ans = 0; k9m = 1101101; p2d = 0; while(k9m > 0) { p2d = (p2d + ((k9m % 10) * x7f)); x7f = (x7f * 2); k9m = (k9m / 10); } ans = p2d;
mbpp_24.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, x7f, k9m, p2d; x7f = 1; ans = 0; k9m = 1101101; p2d = 0; while(k9m > 0) { p2d = (p2d + ((k9m % 10) * x7f)); x7f = (x7f * 2); k9m = (k9m / 10); } ans = p2d;
false
true
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null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": 1, "p2d": 45, "x7f": 64, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 14", "Rule 34" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": 1, "p2d": 45, "x7f": 64, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 2", "Rule 2", "Rule 1", "Rule 9", "Rule 2", "Rule 7", "Rule 5", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": 1, "p2d": 109, "x7f": 64, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 9, "rule": [ "Rule 2", "Rule 1", "Rule 7", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": 1, "p2d": 109, "x7f": 128, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 10, "rule": [ "Rule 2", "Rule 1", "Rule 8", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": 0, "p2d": 109, "x7f": 128, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 15", "Rule 35" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": 0, "p2d": 109, "x7f": 128, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 12, "rule": [ "Rule 2", "Rule 4" ], "state": { "ans": 109, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": 0, "p2d": 109, "x7f": 128, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 109, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": 0, "p2d": 109, "x7f": 128, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
109
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <k9m>0</k9m> <p2d>0</p2d> <x7f>0</x7f> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <k9m>0</k9m> <p2d>0</p2d> <x7f>1</x7f> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <k9m>0</k9m> <p2d>0</p2d> <x7f>1</x7f> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <k9m>1101101</k9m> <p2d>0</p2d> <x7f>1</x7f> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k9m>1101101</k9m> <p2d>0</p2d> <x7f>1</x7f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <k9m>1101101</k9m> <p2d>0</p2d> <x7f>1</x7f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <k9m>1101101</k9m> <p2d>1</p2d> <x7f>1</x7f> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <k9m>1101101</k9m> <p2d>1</p2d> <x7f>2</x7f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <k9m>110110</k9m> <p2d>1</p2d> <x7f>2</x7f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <k9m>110110</k9m> <p2d>1</p2d> <x7f>2</x7f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <k9m>110110</k9m> <p2d>1</p2d> <x7f>2</x7f> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <k9m>110110</k9m> <p2d>1</p2d> <x7f>4</x7f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <k9m>11011</k9m> <p2d>1</p2d> <x7f>4</x7f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <k9m>11011</k9m> <p2d>1</p2d> <x7f>4</x7f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <k9m>11011</k9m> <p2d>5</p2d> <x7f>4</x7f> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <k9m>11011</k9m> <p2d>5</p2d> <x7f>8</x7f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <k9m>1101</k9m> <p2d>5</p2d> <x7f>8</x7f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <k9m>1101</k9m> <p2d>5</p2d> <x7f>8</x7f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <k9m>1101</k9m> <p2d>13</p2d> <x7f>8</x7f> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <k9m>1101</k9m> <p2d>13</p2d> <x7f>16</x7f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <k9m>110</k9m> <p2d>13</p2d> <x7f>16</x7f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <k9m>110</k9m> <p2d>13</p2d> <x7f>16</x7f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <k9m>110</k9m> <p2d>13</p2d> <x7f>16</x7f> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <k9m>110</k9m> <p2d>13</p2d> <x7f>32</x7f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <k9m>11</k9m> <p2d>13</p2d> <x7f>32</x7f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <k9m>11</k9m> <p2d>13</p2d> <x7f>32</x7f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <k9m>11</k9m> <p2d>45</p2d> <x7f>32</x7f> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <k9m>11</k9m> <p2d>45</p2d> <x7f>64</x7f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <k9m>1</k9m> <p2d>45</p2d> <x7f>64</x7f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <k9m>1</k9m> <p2d>45</p2d> <x7f>64</x7f> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <k9m>1</k9m> <p2d>109</p2d> <x7f>64</x7f> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <k9m>1</k9m> <p2d>109</p2d> <x7f>128</x7f> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <k9m>0</k9m> <p2d>109</p2d> <x7f>128</x7f> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <k9m>0</k9m> <p2d>109</p2d> <x7f>128</x7f> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>109</ans> <k9m>0</k9m> <p2d>109</p2d> <x7f>128</x7f> </program_state> </step> </ans>
etp-uk-SOS-mbpp_244.imp
int ans, n, m; n = 35; if(n <= 3) { ans = 4; halt; } m = 2; while((m * m) <= n) { m = (m + 1); } ans = (m * m);
mbpp_244.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n, m; n = 35; if(n <= 3) { ans = 4; halt; } m = 2; while((m * m) <= n) { m = (m + 1); } ans = (m * m);
false
true
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"y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 9, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 13", "Rule 35" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 6, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 35, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 13, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 36, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 6, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 35, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 36, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 6, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 35, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
36
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <m>0</m> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <m>0</m> <n>35</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <m>0</m> <n>35</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <m>2</m> <n>35</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <m>2</m> <n>35</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <m>3</m> <n>35</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <m>3</m> <n>35</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <m>4</m> <n>35</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <m>4</m> <n>35</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <m>5</m> <n>35</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <m>5</m> <n>35</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <m>6</m> <n>35</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <m>6</m> <n>35</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>36</ans> <m>6</m> <n>35</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_271.imp
int ans, i, n, j; i = 1; n = 3; while(i <= n) { j = (2 * i); ans = (ans + ((((j * j) * j) * j) * j)); i = (i + 1); }
mbpp_271.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, n, j; i = 1; n = 3; while(i <= n) { j = (2 * i); ans = (ans + ((((j * j) * j) * j) * j)); i = (i + 1); }
false
true
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"k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 12", "Rule 34" ], "state": { "ans": 1056, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 4, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 3, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, 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null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 7", "Rule 2", "Rule 7", "Rule 2", "Rule 7", "Rule 5", "Rule 4" ], "state": { "ans": 8832, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 6, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 3, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": 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null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 8832, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 6, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 3, "y": null, "xyz": null, "i": 4, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
8,832
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>0</j> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>0</j> <n>3</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>0</j> <n>3</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>1</i> <j>2</j> <n>3</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>32</ans> <i>1</i> <j>2</j> <n>3</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>32</ans> <i>2</i> <j>2</j> <n>3</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>32</ans> <i>2</i> <j>2</j> <n>3</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>32</ans> <i>2</i> <j>4</j> <n>3</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1056</ans> <i>2</i> <j>4</j> <n>3</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>1056</ans> <i>3</i> <j>4</j> <n>3</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1056</ans> <i>3</i> <j>4</j> <n>3</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1056</ans> <i>3</i> <j>6</j> <n>3</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>8832</ans> <i>3</i> <j>6</j> <n>3</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>8832</ans> <i>4</i> <j>6</j> <n>3</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>8832</ans> <i>4</i> <j>6</j> <n>3</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_283.imp
int cnt, ans, n; n = 51241; while(n > 0) { cnt = (cnt + (n % 10)); n = (n / 10); } if((cnt % 2) == 0) { ans = 1; }
mbpp_283.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int cnt, ans, n; n = 51241; while(n > 0) { cnt = (cnt + (n % 10)); n = (n / 10); } if((cnt % 2) == 0) { ans = 1; }
false
true
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0
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <cnt>0</cnt> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <cnt>0</cnt> <n>51241</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <cnt>0</cnt> <n>51241</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <cnt>1</cnt> <n>51241</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <cnt>1</cnt> <n>5124</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <cnt>1</cnt> <n>5124</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <cnt>5</cnt> <n>5124</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <cnt>5</cnt> <n>512</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <cnt>5</cnt> <n>512</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <cnt>7</cnt> <n>512</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <cnt>7</cnt> <n>51</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <cnt>7</cnt> <n>51</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <cnt>8</cnt> <n>51</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <cnt>8</cnt> <n>5</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <cnt>8</cnt> <n>5</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <cnt>13</cnt> <n>5</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <cnt>13</cnt> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <cnt>13</cnt> <n>0</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <cnt>13</cnt> <n>0</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_295.imp
int ans, i, n; i = 1; n = 12; while(i < n) { if((n % i) == 0) { ans = (ans + i); } i = (i + 1); }
mbpp_295.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, n; i = 1; n = 12; while(i < n) { if((n % i) == 0) { ans = (ans + i); } i = (i + 1); }
false
true
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"vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 10", "Rule 34" ], "state": { "ans": 16, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 12, "y": null, "xyz": null, "i": 10, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 9", "Rule 1", "Rule 19", "Rule 30" ], "state": { "ans": 16, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 12, "y": null, "xyz": null, "i": 10, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 10, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 16, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 12, "y": null, "xyz": null, "i": 11, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 10", "Rule 34" ], "state": { "ans": 16, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 12, "y": null, "xyz": null, "i": 11, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 9", "Rule 1", "Rule 19", "Rule 30" ], "state": { "ans": 16, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 12, "y": null, "xyz": null, "i": 11, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 10, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 16, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 12, "y": null, "xyz": null, "i": 12, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 11", "Rule 35" ], "state": { "ans": 16, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 12, "y": null, "xyz": null, "i": 12, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 16, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 12, "y": null, "xyz": null, "i": 12, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
16
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>12</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>1</ans> <i>1</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1</ans> <i>2</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <i>2</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <i>2</i> <n>12</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>3</ans> <i>2</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>3</ans> <i>3</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>3</ans> <i>3</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>3</ans> <i>3</i> <n>12</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>6</ans> <i>3</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>6</ans> <i>4</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>6</ans> <i>4</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>6</ans> <i>4</i> <n>12</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>10</ans> <i>4</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>10</ans> <i>5</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>10</ans> <i>5</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>10</ans> <i>5</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>10</ans> <i>6</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>10</ans> <i>6</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>10</ans> <i>6</i> <n>12</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>16</ans> <i>6</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>16</ans> <i>7</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>16</ans> <i>7</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>16</ans> <i>7</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>16</ans> <i>8</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>16</ans> <i>8</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>16</ans> <i>8</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>16</ans> <i>9</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>16</ans> <i>9</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>16</ans> <i>9</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>16</ans> <i>10</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>16</ans> <i>10</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>16</ans> <i>10</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>16</ans> <i>11</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>16</ans> <i>11</i> <n>12</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>16</ans> <i>11</i> <n>12</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>16</ans> <i>12</i> <n>12</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>16</ans> <i>12</i> <n>12</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_318.imp
int ans, i, s, k, j, vol; s = 2; i = 1; while(i <= s) { j = 1; while(j <= s) { k = 1; while(k <= s) { if(((i + j) + k) > s) { k = (k + 1); continue; } vol = ((i * j) * k); if(vol > ans) { ans = vol; } k = (k + 1); } j = (j + 1); } i = (i + 1); }
mbpp_318.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, s, k, j, vol; s = 2; i = 1; while(i <= s) { j = 1; while(j <= s) { k = 1; while(k <= s) { if(((i + j) + k) > s) { k = (k + 1); continue; } vol = ((i * j) * k); if(vol > ans) { ans = vol; } k = (k + 1); } j = (j + 1); } i = (i + 1); }
false
true
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<linenumber>12</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <k>1</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <k>2</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <k>2</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <k>2</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <k>2</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>1</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>1</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>1</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>1</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>2</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>2</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>2</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>2</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>2</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>24</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>3</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>2</i> <j>3</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>26</linenumber> <program_state> <ans>0</ans> <i>3</i> <j>3</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>3</i> <j>3</j> <k>3</k> <s>2</s> <vol>0</vol> </program_state> </step> </ans>
etp-uk-SOS-mbpp_32.imp
int x, p, ans, k; p = 2; x = 3; k = 15; while(x < k) { if((k % x) == 0) { if(x > p) { p = x; } } x = (x + 1); } ans = p;
mbpp_32.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int x, p, ans, k; p = 2; x = 3; k = 15; while(x < k) { if((k % x) == 0) { if(x > p) { p = x; } } x = (x + 1); } ans = p;
false
true
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5
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<k>15</k> <p>3</p> <x>4</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>3</p> <x>4</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>3</p> <x>4</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>3</p> <x>5</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>3</p> <x>5</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>3</p> <x>5</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>3</p> <x>5</x> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>5</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>6</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>6</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>6</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>7</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>7</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>7</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>8</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>8</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>8</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>9</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>9</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>9</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>10</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>10</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>10</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>11</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>11</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>11</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>12</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>12</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>12</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>13</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>13</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>13</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>14</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>14</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>14</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>15</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <k>15</k> <p>5</p> <x>15</x> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>5</ans> <k>15</k> <p>5</p> <x>15</x> </program_state> </step> </ans>
etp-uk-SOS-mbpp_320.imp
int a, b, n, ans, i, j; n = 6; i = 1; while(i <= n) { a = (a + i); i = (i + 1); } j = 1; while(j <= n) { b = (b + (j * j)); j = (j + 1); } ans = ((a * a) - b);
mbpp_320.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int a, b, n, ans, i, j; n = 6; i = 1; while(i <= n) { a = (a + i); i = (i + 1); } j = 1; while(j <= n) { b = (b + (j * j)); j = (j + 1); } ans = ((a * a) - b);
false
true
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350
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<a>21</a> <ans>0</ans> <b>0</b> <i>7</i> <j>1</j> <n>6</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>0</b> <i>7</i> <j>1</j> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>1</b> <i>7</i> <j>1</j> <n>6</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>1</b> <i>7</i> <j>2</j> <n>6</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>1</b> <i>7</i> <j>2</j> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>5</b> <i>7</i> <j>2</j> <n>6</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>5</b> <i>7</i> <j>3</j> <n>6</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>5</b> <i>7</i> <j>3</j> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>14</b> <i>7</i> <j>3</j> <n>6</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>14</b> <i>7</i> <j>4</j> <n>6</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>14</b> <i>7</i> <j>4</j> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>30</b> <i>7</i> <j>4</j> <n>6</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>30</b> <i>7</i> <j>5</j> <n>6</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>30</b> <i>7</i> <j>5</j> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>55</b> <i>7</i> <j>5</j> <n>6</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>55</b> <i>7</i> <j>6</j> <n>6</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>55</b> <i>7</i> <j>6</j> <n>6</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>91</b> <i>7</i> <j>6</j> <n>6</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>91</b> <i>7</i> <j>7</j> <n>6</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>21</a> <ans>0</ans> <b>91</b> <i>7</i> <j>7</j> <n>6</n> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <a>21</a> <ans>350</ans> <b>91</b> <i>7</i> <j>7</j> <n>6</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_327.imp
int ans, x, y, z; x = 6; y = 16; z = 20; if((((x != y) || (((z != z) && (x != x))))) || (((y != z) && (y != y)))) { halt; } ans = 1;
mbpp_327.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, x, y, z; x = 6; y = 16; z = 20; if((((x != y) || (((z != z) && (x != x))))) || (((y != z) && (y != y)))) { halt; } ans = 1;
false
true
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null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 6, "B": null, "X": null, "n": null, "y": 16, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": 0, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 6, "B": null, "X": null, "n": null, "y": 16, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": 20, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 2", "Rule 20", "Rule 2", "Rule 2", "Rule 21", "Rule 2", "Rule 2", "Rule 21", "Rule 23", "Rule 24", "Rule 2", "Rule 2", "Rule 20", "Rule 2", "Rule 2", "Rule 21", "Rule 23", "Rule 24", "Rule 31" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 6, "B": null, "X": null, "n": null, "y": 16, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": 20, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 41" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 6, "B": null, "X": null, "n": null, "y": 16, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": 20, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 6, "B": null, "X": null, "n": null, "y": 16, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": 20, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
0
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <x>0</x> <y>0</y> <z>0</z> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <x>6</x> <y>0</y> <z>0</z> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <x>6</x> <y>16</y> <z>0</z> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>6</x> <y>16</y> <z>20</z> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <x>6</x> <y>16</y> <z>20</z> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <x>6</x> <y>16</y> <z>20</z> </program_state> </step> </ans>
etp-uk-SOS-mbpp_334.imp
int ans, a, b, c; a = 7; b = 10; c = 5; if(((((a + b)) > c) && (((b + c) > a))) && ((a + c) > b)) { ans = 1; }
mbpp_334.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, a, b, c; a = 7; b = 10; c = 5; if(((((a + b)) > c) && (((b + c) > a))) && ((a + c) > b)) { ans = 1; }
false
true
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1
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>7</a> <ans>0</ans> <b>0</b> <c>0</c> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>7</a> <ans>0</ans> <b>10</b> <c>0</c> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>7</a> <ans>0</ans> <b>10</b> <c>5</c> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>7</a> <ans>0</ans> <b>10</b> <c>5</c> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>7</a> <ans>1</ans> <b>10</b> <c>5</c> </program_state> </step> </ans>
etp-uk-SOS-mbpp_339.imp
int ans, x, y, i; x = 5; y = 10; if(x == y) { ans = x; halt; } else { i = 1; while(((x % i) == 0) && ((y % i) == 0)) { i = (i + 1); } ans = i; }
mbpp_339.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, x, y, i; x = 5; y = 10; if(x == y) { ans = x; halt; } else { i = 1; while(((x % i) == 0) && ((y % i) == 0)) { i = (i + 1); } ans = i; }
false
true
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10, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 16, "rule": [ "Rule 2", "Rule 4" ], "state": { "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 5, "B": null, "X": null, "n": null, "y": 10, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 2, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 5, "B": null, "X": null, "n": null, "y": 10, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
2
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <x>5</x> <y>0</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <x>5</x> <y>10</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>0</i> <x>5</x> <y>10</y> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>1</i> <x>5</x> <y>10</y> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>0</ans> <i>1</i> <x>5</x> <y>10</y> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <i>2</i> <x>5</x> <y>10</y> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>0</ans> <i>2</i> <x>5</x> <y>10</y> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>2</ans> <i>2</i> <x>5</x> <y>10</y> </program_state> </step> </ans>
etp-uk-SOS-mbpp_346.imp
int ans, n, k; n = 4; k = 3; while(n > 0) { if(k == 0) { halt; } k = (k - 1); if((n % 2) != 0) { n = (n + 1); ans = (ans + 1); } else { n = (n / 2); ans = (ans + 2); } }
mbpp_346.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n, k; n = 4; k = 3; while(n > 0) { if(k == 0) { halt; } k = (k - 1); if((n % 2) != 0) { n = (n + 1); ans = (ans + 1); } else { n = (n / 2); ans = (ans + 2); } }
false
true
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null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 11, "rule": [ "Rule 2", "Rule 1", "Rule 9", "Rule 1", "Rule 20", "Rule 33" ], "state": { "ans": 4, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": 0, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 13, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 4, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": 0, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 2, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 14, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": 0, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 2, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 1", "Rule 14", "Rule 34" ], "state": { "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": 0, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 2, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 18", "Rule 31" ], "state": { "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": 0, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 2, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 41" ], "state": { "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": 0, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 2, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": 0, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 2, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
5
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <k>0</k> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <k>0</k> <n>4</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <k>3</k> <n>4</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <k>3</k> <n>4</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <k>3</k> <n>4</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <k>2</k> <n>4</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <k>2</k> <n>4</n> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <k>2</k> <n>2</n> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>2</ans> <k>2</k> <n>2</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>2</ans> <k>2</k> <n>2</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>2</ans> <k>2</k> <n>2</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>2</ans> <k>1</k> <n>2</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>2</ans> <k>1</k> <n>2</n> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>2</ans> <k>1</k> <n>1</n> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>4</ans> <k>1</k> <n>1</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>4</ans> <k>1</k> <n>1</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>4</ans> <k>1</k> <n>1</n> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>4</ans> <k>0</k> <n>1</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>4</ans> <k>0</k> <n>1</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>4</ans> <k>0</k> <n>2</n> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>5</ans> <k>0</k> <n>2</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>5</ans> <k>0</k> <n>2</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>5</ans> <k>0</k> <n>2</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>5</ans> <k>0</k> <n>2</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_35.imp
int ans, n; n = 6; ans = (n * (n + 1));
mbpp_35.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n; n = 6; ans = (n * (n + 1));
false
true
[ { "line_number": 1, "rule": [ "Rule 38", "Rule 3", "Rule 3" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 2, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 2", "Rule 1", "Rule 5", "Rule 7", "Rule 4" ], "state": { "ans": 42, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 42, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
42
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n>6</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>42</ans> <n>6</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_354.imp
int ans, i, a, d, n; a = 2; n = 6; d = 4; while(i < n) { ans = (a + (i * d)); i = (i + 1); }
mbpp_354.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, a, d, n; a = 2; n = 6; d = 4; while(i < n) { ans = (a + (i * d)); i = (i + 1); }
false
true
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22
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <d>0</d> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>2</a> <ans>0</ans> <d>0</d> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>2</a> <ans>0</ans> <d>0</d> <i>0</i> <n>6</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>2</a> <ans>0</ans> <d>4</d> <i>0</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>2</a> <ans>0</ans> <d>4</d> <i>0</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>2</a> <ans>2</ans> <d>4</d> <i>0</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>2</a> <ans>2</ans> <d>4</d> <i>1</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>2</a> <ans>2</ans> <d>4</d> <i>1</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>2</a> <ans>6</ans> <d>4</d> <i>1</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>2</a> <ans>6</ans> <d>4</d> <i>2</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>2</a> <ans>6</ans> <d>4</d> <i>2</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>2</a> <ans>10</ans> <d>4</d> <i>2</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>2</a> <ans>10</ans> <d>4</d> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>2</a> <ans>10</ans> <d>4</d> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>2</a> <ans>14</ans> <d>4</d> <i>3</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>2</a> <ans>14</ans> <d>4</d> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>2</a> <ans>14</ans> <d>4</d> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>2</a> <ans>18</ans> <d>4</d> <i>4</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>2</a> <ans>18</ans> <d>4</d> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>2</a> <ans>18</ans> <d>4</d> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>2</a> <ans>22</ans> <d>4</d> <i>5</i> <n>6</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>2</a> <ans>22</ans> <d>4</d> <i>6</i> <n>6</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>2</a> <ans>22</ans> <d>4</d> <i>6</i> <n>6</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_355.imp
int ans, x, y; y = 2; while(x < ((y * y) * y)) { ans = (ans + 1); x = (x + 1); }
mbpp_355.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, x, y; y = 2; while(x < ((y * y) * y)) { ans = (ans + 1); x = (x + 1); }
false
true
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"r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 7", "Rule 10", "Rule 34" ], "state": { "ans": 7, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 7, "B": null, "X": null, "n": null, "y": 2, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 7, "B": null, "X": null, "n": null, "y": 2, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 8, "B": null, "X": null, "n": null, "y": 2, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 7", "Rule 11", "Rule 35" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 8, "B": null, "X": null, "n": null, "y": 2, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 8, "B": null, "X": null, "n": null, "y": 2, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
8
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <x>0</x> <y>2</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <x>0</x> <y>2</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>1</ans> <x>0</x> <y>2</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <x>1</x> <y>2</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>1</ans> <x>1</x> <y>2</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>2</ans> <x>1</x> <y>2</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>2</ans> <x>2</x> <y>2</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>2</ans> <x>2</x> <y>2</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>3</ans> <x>2</x> <y>2</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>3</ans> <x>3</x> <y>2</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>3</ans> <x>3</x> <y>2</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>4</ans> <x>3</x> <y>2</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>4</ans> <x>4</x> <y>2</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>4</ans> <x>4</x> <y>2</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>5</ans> <x>4</x> <y>2</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>5</ans> <x>5</x> <y>2</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>5</ans> <x>5</x> <y>2</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>6</ans> <x>5</x> <y>2</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>6</ans> <x>6</x> <y>2</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>6</ans> <x>6</x> <y>2</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>7</ans> <x>6</x> <y>2</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>7</ans> <x>7</x> <y>2</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>7</ans> <x>7</x> <y>2</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>8</ans> <x>7</x> <y>2</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>8</ans> <x>8</x> <y>2</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>8</ans> <x>8</x> <y>2</y> </program_state> </step> </ans>
etp-uk-SOS-mbpp_356.imp
int ans, a, b; a = 45; b = 95; ans = ((a + b) % 180); if(ans > 90) { ans = (180 - ans); }
mbpp_356.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, a, b; a = 45; b = 95; ans = ((a + b) % 180); if(ans > 90) { ans = (180 - ans); }
false
true
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"x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 1", "Rule 2", "Rule 6", "Rule 4" ], "state": { "ans": 40, "num": null, "a": 45, "b": 95, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 40, "num": null, "a": 45, "b": 95, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
40
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>45</a> <ans>0</ans> <b>0</b> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>45</a> <ans>0</ans> <b>95</b> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>45</a> <ans>140</ans> <b>95</b> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>45</a> <ans>140</ans> <b>95</b> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>45</a> <ans>40</ans> <b>95</b> </program_state> </step> </ans>
etp-uk-SOS-mbpp_36.imp
int x, ans, k, m, i; x = 0; k = 5; m = 6; i = 5; while(i > 0) { i = (i - 1); k = (k * 10); x = (k / m); k = (k % m); } ans = x;
mbpp_36.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int x, ans, k, m, i; x = 0; k = 5; m = 6; i = 5; while(i > 0) { i = (i - 1); k = (k * 10); x = (k / m); k = (k % m); } ans = x;
false
true
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3
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>0</k> <m>0</m> <x>0</x> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>0</k> <m>0</m> <x>0</x> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>5</k> <m>0</m> <x>0</x> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>5</k> <m>6</m> <x>0</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>5</i> <k>5</k> <m>6</m> <x>0</x> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>5</i> <k>5</k> <m>6</m> <x>0</x> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>4</i> <k>5</k> <m>6</m> <x>0</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>4</i> <k>50</k> <m>6</m> <x>0</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>4</i> <k>50</k> <m>6</m> <x>8</x> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>4</i> <k>2</k> <m>6</m> <x>8</x> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>4</i> <k>2</k> <m>6</m> <x>8</x> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>3</i> <k>2</k> <m>6</m> <x>8</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>3</i> <k>20</k> <m>6</m> <x>8</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>3</i> <k>20</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>3</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>3</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>2</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>2</i> <k>20</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>2</i> <k>20</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>2</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>2</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>1</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>1</i> <k>20</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>1</i> <k>20</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>1</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>1</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>20</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>20</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>3</ans> <i>0</i> <k>2</k> <m>6</m> <x>3</x> </program_state> </step> </ans>
etp-uk-SOS-mbpp_365.imp
int ans, n; n = 11223305; while(n > 0) { n = (n / 10); ans = (ans + 1); }
mbpp_365.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n; n = 11223305; while(n > 0) { n = (n / 10); ans = (ans + 1); }
false
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"r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 1", "Rule 14", "Rule 34" ], "state": { "ans": 6, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 11, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 1", "Rule 8", "Rule 4" ], "state": { "ans": 6, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 7, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 1", "Rule 14", "Rule 34" ], "state": { "ans": 7, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 1", "Rule 8", "Rule 4" ], "state": { "ans": 7, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 1", "Rule 15", "Rule 35" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
8
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n>11223305</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <n>11223305</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <n>1122330</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <n>1122330</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>1</ans> <n>1122330</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>1</ans> <n>112233</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>2</ans> <n>112233</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>2</ans> <n>112233</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>2</ans> <n>11223</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>3</ans> <n>11223</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>3</ans> <n>11223</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>3</ans> <n>1122</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>4</ans> <n>1122</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>4</ans> <n>1122</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>4</ans> <n>112</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>5</ans> <n>112</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>5</ans> <n>112</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>5</ans> <n>11</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>6</ans> <n>11</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>6</ans> <n>11</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>6</ans> <n>1</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>7</ans> <n>1</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>7</ans> <n>1</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>7</ans> <n>0</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>8</ans> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>8</ans> <n>0</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_375.imp
int i, ans, n, m; n = 1111; m = 5; while(i < m) { ans = (ans + (n / m)); i = (i + 1); }
mbpp_375.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int i, ans, n, m; n = 1111; m = 5; while(i < m) { ans = (ans + (n / m)); i = (i + 1); }
false
true
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"aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 2", "Rule 8", "Rule 5", "Rule 4" ], "state": { "ans": 666, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 5, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1111, "y": null, "xyz": null, "i": 2, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 666, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 5, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1111, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, 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"n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 2", "Rule 8", "Rule 5", "Rule 4" ], "state": { "ans": 888, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 5, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1111, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 888, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 5, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1111, "y": null, "xyz": null, "i": 4, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, 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"bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 2", "Rule 8", "Rule 5", "Rule 4" ], "state": { "ans": 1110, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 5, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1111, "y": null, "xyz": null, "i": 4, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 1110, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 5, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1111, "y": null, "xyz": null, "i": 5, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 11", "Rule 35" ], "state": { "ans": 1110, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 5, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1111, "y": null, "xyz": null, "i": 5, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 1110, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": 5, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1111, "y": null, "xyz": null, "i": 5, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
1,110
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <m>0</m> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <m>0</m> <n>1111</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>0</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>222</ans> <i>0</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>222</ans> <i>1</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>222</ans> <i>1</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>444</ans> <i>1</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>444</ans> <i>2</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>444</ans> <i>2</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>666</ans> <i>2</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>666</ans> <i>3</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>666</ans> <i>3</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>888</ans> <i>3</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>888</ans> <i>4</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>888</ans> <i>4</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1110</ans> <i>4</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1110</ans> <i>5</i> <m>5</m> <n>1111</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1110</ans> <i>5</i> <m>5</m> <n>1111</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_402.imp
int ans, i, r, p, n; n = 15; r = 12; p = 43; ans = 1; i = 1; while(i <= r) { ans = ((ans * ((n - i) + 1)) / i); i = (i + 1); } ans = (ans % p);
mbpp_402.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, r, p, n; n = 15; r = 12; p = 43; ans = 1; i = 1; while(i <= r) { ans = ((ans * ((n - i) + 1)) / i); i = (i + 1); } ans = (ans % p);
false
true
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{ "ans": 25, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": 43, "q": null, "r": 12, "x": null, "B": null, "X": null, "n": 15, "y": null, "xyz": null, "i": 13, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
25
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> <p>0</p> <r>0</r> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>15</n> <p>0</p> <r>0</r> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>15</n> <p>0</p> <r>12</r> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>1</ans> <i>0</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <i>1</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1</ans> <i>1</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>15</ans> <i>1</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>15</ans> <i>2</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>15</ans> <i>2</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>105</ans> <i>2</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>105</ans> <i>3</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>105</ans> <i>3</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>455</ans> <i>3</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>455</ans> <i>4</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>455</ans> <i>4</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>1365</ans> <i>4</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1365</ans> <i>5</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1365</ans> <i>5</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>3003</ans> <i>5</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>3003</ans> <i>6</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>3003</ans> <i>6</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>5005</ans> <i>6</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>5005</ans> <i>7</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>5005</ans> <i>7</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>6435</ans> <i>7</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>6435</ans> <i>8</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>6435</ans> <i>8</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>6435</ans> <i>8</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>6435</ans> <i>9</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>6435</ans> <i>9</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>5005</ans> <i>9</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>5005</ans> <i>10</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>5005</ans> <i>10</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>3003</ans> <i>10</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>3003</ans> <i>11</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>3003</ans> <i>11</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>1365</ans> <i>11</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1365</ans> <i>12</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1365</ans> <i>12</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>455</ans> <i>12</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>455</ans> <i>13</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>455</ans> <i>13</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>25</ans> <i>13</i> <n>15</n> <p>43</p> <r>12</r> </program_state> </step> </ans>
etp-uk-SOS-mbpp_420.imp
int n, ans, i; n = 4; i = 1; while(i <= n) { ans = (ans + (((2 * i) * (2 * i)) * (2 * i))); i = (i + 1); }
mbpp_420.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int n, ans, i; n = 4; i = 1; while(i <= n) { ans = (ans + (((2 * i) * (2 * i)) * (2 * i))); i = (i + 1); }
false
true
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{ "ans": 800, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": 5, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
800
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>4</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>4</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>1</i> <n>4</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>8</ans> <i>1</i> <n>4</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>8</ans> <i>2</i> <n>4</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>8</ans> <i>2</i> <n>4</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>72</ans> <i>2</i> <n>4</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>72</ans> <i>3</i> <n>4</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>72</ans> <i>3</i> <n>4</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>288</ans> <i>3</i> <n>4</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>288</ans> <i>4</i> <n>4</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>288</ans> <i>4</i> <n>4</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>800</ans> <i>4</i> <n>4</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>800</ans> <i>5</i> <n>4</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>800</ans> <i>5</i> <n>4</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_430.imp
int ans, a, b, c, x, y; a = 5; b = 3; c = 2; x = (b - a); y = (c - b); if(x > y) { ans = (- 2336); halt; } else { if(x < y) { ans = (- 198); } else { ans = (- 130); } }
mbpp_430.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, a, b, c, x, y; a = 5; b = 3; c = 2; x = (b - a); y = (c - b); if(x > y) { ans = (- 2336); halt; } else { if(x < y) { ans = (- 198); } else { ans = (- 130); } }
false
true
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"i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 6", "Rule 4" ], "state": { "ans": 0, "num": null, "a": 5, "b": 3, "c": 2, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": -2, "B": null, "X": null, "n": null, "y": -1, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 2", "Rule 15", "Rule 32" ], "state": { "ans": 0, "num": null, "a": 5, "b": 3, "c": 2, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": -2, "B": null, "X": null, "n": null, "y": -1, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 14, "rule": [ "Rule 2", "Rule 2", "Rule 10", "Rule 33" ], "state": { "ans": 0, "num": null, "a": 5, "b": 3, "c": 2, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": -2, "B": null, "X": null, "n": null, "y": -1, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 16, "rule": [ "Rule 1", "Rule 48", "Rule 4" ], "state": { "ans": -198, "num": null, "a": 5, "b": 3, "c": 2, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": -2, "B": null, "X": null, "n": null, "y": -1, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": -198, "num": null, "a": 5, "b": 3, "c": 2, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": -2, "B": null, "X": null, "n": null, "y": -1, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
-198
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>0</b> <c>0</c> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>3</b> <c>0</c> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>3</b> <c>2</c> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>3</b> <c>2</c> <x>-2</x> <y>0</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>3</b> <c>2</c> <x>-2</x> <y>-1</y> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>3</b> <c>2</c> <x>-2</x> <y>-1</y> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>5</a> <ans>0</ans> <b>3</b> <c>2</c> <x>-2</x> <y>-1</y> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>5</a> <ans>-198</ans> <b>3</b> <c>2</c> <x>-2</x> <y>-1</y> </program_state> </step> </ans>
etp-uk-SOS-mbpp_467.imp
int ans, a, i; i = 1; a = 33; while(a > 0) { ans = (ans + ((a % 8) * i)); a = (a / 8); i = (i * 10); }
mbpp_467.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, a, i; i = 1; a = 33; while(a > 0) { ans = (ans + ((a % 8) * i)); a = (a / 8); i = (i * 10); }
false
true
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{ "ans": 41, "num": null, "a": 0, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": 100, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
41
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <i>0</i> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <i>1</i> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>33</a> <ans>0</ans> <i>1</i> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>33</a> <ans>0</ans> <i>1</i> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>33</a> <ans>1</ans> <i>1</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>4</a> <ans>1</ans> <i>1</i> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>4</a> <ans>1</ans> <i>10</i> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>4</a> <ans>1</ans> <i>10</i> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>4</a> <ans>41</ans> <i>10</i> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>0</a> <ans>41</ans> <i>10</i> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>41</ans> <i>100</i> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>0</a> <ans>41</ans> <i>100</i> </program_state> </step> </ans>
etp-uk-SOS-mbpp_47.imp
int xyz, ans, p, q, j; xyz = 1; p = 6; q = 8; if(p == q) { ans = 1; halt; } if((q - p) >= 5) { ans = 0; halt; } j = (p + 1); while(j <= q) { xyz = ((xyz * (j % 10)) % 10); j = (j + 1); } ans = (xyz % 10);
mbpp_47.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int xyz, ans, p, q, j; xyz = 1; p = 6; q = 8; if(p == q) { ans = 1; halt; } if((q - p) >= 5) { ans = 0; halt; } j = (p + 1); while(j <= q) { xyz = ((xyz * (j % 10)) % 10); j = (j + 1); } ans = (xyz % 10);
false
true
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{ "ans": 6, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 9, "k": null, "m": null, "p": 6, "q": 8, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": 6, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
6
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <j>0</j> <p>0</p> <q>0</q> <xyz>0</xyz> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <j>0</j> <p>0</p> <q>0</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <j>0</j> <p>6</p> <q>0</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <j>0</j> <p>6</p> <q>8</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <j>0</j> <p>6</p> <q>8</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <j>0</j> <p>6</p> <q>8</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <ans>0</ans> <j>7</j> <p>6</p> <q>8</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <j>7</j> <p>6</p> <q>8</q> <xyz>1</xyz> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <j>7</j> <p>6</p> <q>8</q> <xyz>7</xyz> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>0</ans> <j>8</j> <p>6</p> <q>8</q> <xyz>7</xyz> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <j>8</j> <p>6</p> <q>8</q> <xyz>7</xyz> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <ans>0</ans> <j>8</j> <p>6</p> <q>8</q> <xyz>6</xyz> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>0</ans> <j>9</j> <p>6</p> <q>8</q> <xyz>6</xyz> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>0</ans> <j>9</j> <p>6</p> <q>8</q> <xyz>6</xyz> </program_state> </step> <step> <linenumber>21</linenumber> <program_state> <ans>6</ans> <j>9</j> <p>6</p> <q>8</q> <xyz>6</xyz> </program_state> </step> </ans>
etp-uk-SOS-mbpp_479.imp
int ans, n; n = 456; while(n > 0) { ans = (n % 10); n = (n / 10); }
mbpp_479.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n; n = 456; while(n > 0) { ans = (n % 10); n = (n / 10); }
false
true
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"t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 8", "Rule 4" ], "state": { "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 1", "Rule 14", "Rule 34" ], "state": { "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 1", "Rule 9", "Rule 4" ], "state": { "ans": 4, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 4, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 8", "Rule 4" ], "state": { "ans": 4, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 1", "Rule 15", "Rule 35" ], "state": { "ans": 4, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 4, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
4
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n>456</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <n>456</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>6</ans> <n>456</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>6</ans> <n>45</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>6</ans> <n>45</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>5</ans> <n>45</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>5</ans> <n>4</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>5</ans> <n>4</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>4</ans> <n>4</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>4</ans> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>4</ans> <n>0</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_483.imp
int ans, x, y, i; x = 15; y = 1; i = 1; while(y < x) { y = (y * i); i = (i + 1); } ans = i;
mbpp_483.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, x, y, i; x = 15; y = 1; i = 1; while(y < x) { y = (y * i); i = (i + 1); } ans = i;
false
true
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null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 15, "B": null, "X": null, "n": null, "y": 6, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 15, "B": null, "X": null, "n": null, "y": 6, "xyz": null, "i": 4, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 2", "Rule 10", "Rule 34" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 15, "B": null, "X": null, "n": null, "y": 6, "xyz": null, "i": 4, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 15, "B": null, "X": null, "n": null, "y": 24, "xyz": null, "i": 4, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 15, "B": null, "X": null, "n": null, "y": 24, "xyz": null, "i": 5, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 2", "Rule 11", "Rule 35" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 15, "B": null, "X": null, "n": null, "y": 24, "xyz": null, "i": 5, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 10, "rule": [ "Rule 2", "Rule 4" ], "state": { "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 15, "B": null, "X": null, "n": null, "y": 24, "xyz": null, "i": 5, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 5, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 15, "B": null, "X": null, "n": null, "y": 24, "xyz": null, "i": 5, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
5
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <x>15</x> <y>0</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <x>15</x> <y>1</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>1</i> <x>15</x> <y>1</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>1</i> <x>15</x> <y>1</y> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>1</i> <x>15</x> <y>1</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>2</i> <x>15</x> <y>1</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>2</i> <x>15</x> <y>1</y> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>2</i> <x>15</x> <y>2</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>3</i> <x>15</x> <y>2</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>3</i> <x>15</x> <y>2</y> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>3</i> <x>15</x> <y>6</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>4</i> <x>15</x> <y>6</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>4</i> <x>15</x> <y>6</y> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>4</i> <x>15</x> <y>24</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <i>5</i> <x>15</x> <y>24</y> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <i>5</i> <x>15</x> <y>24</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>5</ans> <i>5</i> <x>15</x> <y>24</y> </program_state> </step> </ans>
etp-uk-SOS-mbpp_5.imp
int n, i, ans; n = 9; i = 2; while(i < n) { if((n % i) == 0) { ans = i; halt; } i = (i + 1); } ans = (n - 1);
mbpp_5.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int n, i, ans; n = 9; i = 2; while(i < n) { if((n % i) == 0) { ans = i; halt; } i = (i + 1); } ans = (n - 1);
false
true
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null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 9", "Rule 1", "Rule 18", "Rule 31" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 9, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 2", "Rule 4" ], "state": { "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 9, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 9, "rule": [ "Rule 41" ], "state": { "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 9, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 3, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 9, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
3
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>9</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>9</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>9</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>9</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>9</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>9</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>9</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>3</ans> <i>3</i> <n>9</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>3</ans> <i>3</i> <n>9</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_501.imp
int ans, y, x; x = 12; y = 24; while(y > 0) { if((x % y) == 0) { ans = (ans + 1); } y = (y - 1); }
mbpp_501.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, y, x; x = 12; y = 24; while(y > 0) { if((x % y) == 0) { ans = (ans + 1); } y = (y - 1); }
false
true
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null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 1", "Rule 15", "Rule 35" ], "state": { "ans": 6, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 12, "B": null, "X": null, "n": null, "y": 0, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 6, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 12, "B": null, "X": null, "n": null, "y": 0, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
6
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <x>0</x> <y>0</y> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>0</y> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>24</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>24</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>24</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>23</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>23</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>23</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>22</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>22</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>22</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>21</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>21</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>21</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>20</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>20</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>20</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>19</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>19</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>19</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>18</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>18</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>18</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>17</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>17</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>17</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>16</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>16</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>16</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>15</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>15</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>15</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>14</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>14</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>14</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>13</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>13</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>13</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>12</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>12</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <x>12</x> <y>12</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>12</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>11</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>11</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>11</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>10</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>10</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>10</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>9</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>9</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>9</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>8</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>8</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>8</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>7</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>7</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>7</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>6</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>6</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>1</ans> <x>12</x> <y>6</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>2</ans> <x>12</x> <y>6</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>2</ans> <x>12</x> <y>5</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>2</ans> <x>12</x> <y>5</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>2</ans> <x>12</x> <y>5</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>2</ans> <x>12</x> <y>4</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>2</ans> <x>12</x> <y>4</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>2</ans> <x>12</x> <y>4</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>3</ans> <x>12</x> <y>4</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>3</ans> <x>12</x> <y>3</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>3</ans> <x>12</x> <y>3</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>3</ans> <x>12</x> <y>3</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>4</ans> <x>12</x> <y>3</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>4</ans> <x>12</x> <y>2</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>4</ans> <x>12</x> <y>2</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>4</ans> <x>12</x> <y>2</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>5</ans> <x>12</x> <y>2</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>5</ans> <x>12</x> <y>1</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>5</ans> <x>12</x> <y>1</y> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>5</ans> <x>12</x> <y>1</y> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>6</ans> <x>12</x> <y>1</y> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>6</ans> <x>12</x> <y>0</y> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>6</ans> <x>12</x> <y>0</y> </program_state> </step> </ans>
etp-uk-SOS-mbpp_502.imp
int i, j, ans, n, m; n = 3; m = 3; while(i < n) { j = 0; while(j < m) { if(i == j) { ans = (((n - i) - 1) % 2); halt; } j = (j + 1); } i = (i + 1); } ans = 1;
mbpp_502.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int i, j, ans, n, m; n = 3; m = 3; while(i < n) { j = 0; while(j < m) { if(i == j) { ans = (((n - i) - 1) % 2); halt; } j = (j + 1); } i = (i + 1); } ans = 1;
false
true
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"sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 12, "rule": [ "Rule 41" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 0, "k": null, "m": 3, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 3, "y": null, "xyz": null, "i": 0, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": 0, "k": null, "m": 3, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 3, "y": null, "xyz": null, "i": 0, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
0
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <m>0</m> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <m>0</m> <n>3</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <m>3</m> <n>3</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <m>3</m> <n>3</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <m>3</m> <n>3</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <m>3</m> <n>3</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <m>3</m> <n>3</n> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <m>3</m> <n>3</n> </program_state> </step> <step> <linenumber>12</linenumber> <program_state> <ans>0</ans> <i>0</i> <j>0</j> <m>3</m> <n>3</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_506.imp
int ans, i, k, n; n = 10; k = 3; ans = 1; while(i < k) { ans = (ans * (n - i)); i = (i + 1); }
mbpp_506.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, k, n; n = 10; k = 3; ans = 1; while(i < k) { ans = (ans * (n - i)); i = (i + 1); }
false
true
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{ "ans": 720, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": 3, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": 3, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
720
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>0</k> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>0</k> <n>10</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>0</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>1</ans> <i>0</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>1</ans> <i>0</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>10</ans> <i>0</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>10</ans> <i>1</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>10</ans> <i>1</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>90</ans> <i>1</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>90</ans> <i>2</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>90</ans> <i>2</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>720</ans> <i>2</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>720</ans> <i>3</i> <k>3</k> <n>10</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>720</ans> <i>3</i> <k>3</k> <n>10</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_509.imp
int ans, i, n; n = 11; i = 1; while(i <= n) { ans = (ans + i); i = (i + 1); } ans = (ans / n);
mbpp_509.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, n; n = 11; i = 1; while(i <= n) { ans = (ans + i); i = (i + 1); } ans = (ans / n);
false
true
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"by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 66, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 11, "y": null, "xyz": null, "i": 12, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 2", "Rule 13", "Rule 35" ], "state": { "ans": 66, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 11, "y": null, "xyz": null, "i": 12, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 9, "rule": [ "Rule 2", "Rule 2", "Rule 8", "Rule 4" ], "state": { "ans": 6, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 11, "y": null, "xyz": null, "i": 12, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 6, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 11, "y": null, "xyz": null, "i": 12, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
6
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etp-uk-SOS-mbpp_51.imp
int ans, x, y, z; x = 5; y = 5; z = 5; if((x == y) && (y == z)) { ans = 1; halt; } ans = 0;
mbpp_51.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, x, y, z; x = 5; y = 5; z = 5; if((x == y) && (y == z)) { ans = 1; halt; } ans = 0;
false
true
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1
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <x>0</x> <y>0</y> <z>0</z> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <x>5</x> <y>0</y> <z>0</z> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <x>5</x> <y>5</y> <z>0</z> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>5</x> <y>5</y> <z>5</z> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <x>5</x> <y>5</y> <z>5</z> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1</ans> <x>5</x> <y>5</y> <z>5</z> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>1</ans> <x>5</x> <y>5</y> <z>5</z> </program_state> </step> </ans>
etp-uk-SOS-mbpp_511.imp
int ans, i, n; i = 2; n = 105; while(n > 1) { if((n % i) == 0) { ans = (ans + i); n = (n / i); } else { i = (i + 1); } }
mbpp_511.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, i, n; i = 2; n = 105; while(n > 1) { if((n % i) == 0) { ans = (ans + i); n = (n / i); } else { i = (i + 1); } }
false
true
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"Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 7, "y": null, "xyz": null, "i": 6, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 4, "rule": [ "Rule 2", "Rule 1", "Rule 14", "Rule 34" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 7, "y": null, "xyz": null, "i": 6, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 9", "Rule 1", "Rule 19", "Rule 32" ], "state": { "ans": 8, "num": null, "a": null, 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null, "x": null, "B": null, "X": null, "n": 7, "y": null, "xyz": null, "i": 7, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 2", "Rule 9", "Rule 1", "Rule 18", "Rule 33" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 7, "y": null, "xyz": null, "i": 7, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 2", "Rule 2", "Rule 5", "Rule 4" ], "state": { "ans": 15, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 7, "y": null, "xyz": null, "i": 7, "s": 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{ "ans": 15, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": 7, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
15
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <i>0</i> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>105</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>105</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>2</i> <n>105</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>105</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>105</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <i>3</i> <n>105</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>3</ans> <i>3</i> <n>105</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>3</ans> <i>3</i> <n>35</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>3</ans> <i>3</i> <n>35</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>3</ans> <i>3</i> <n>35</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>3</ans> <i>4</i> <n>35</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>3</ans> <i>4</i> <n>35</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>3</ans> <i>4</i> <n>35</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>3</ans> <i>5</i> <n>35</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>3</ans> <i>5</i> <n>35</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>3</ans> <i>5</i> <n>35</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>8</ans> <i>5</i> <n>35</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>8</ans> <i>5</i> <n>7</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>8</ans> <i>5</i> <n>7</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>8</ans> <i>5</i> <n>7</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>8</ans> <i>6</i> <n>7</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>8</ans> <i>6</i> <n>7</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>8</ans> <i>6</i> <n>7</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>8</ans> <i>7</i> <n>7</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>8</ans> <i>7</i> <n>7</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>8</ans> <i>7</i> <n>7</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>15</ans> <i>7</i> <n>7</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>15</ans> <i>7</i> <n>1</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>15</ans> <i>7</i> <n>1</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_518.imp
int ans, a, b, num, c; num = 400; a = 0; b = num; while(a <= b) { c = (a + ((b - a) / 2)); if((c * c) < num) { a = (c + 1); } else { b = (c - 1); } } ans = a;
mbpp_518.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, a, b, num, c; num = 400; a = 0; b = num; while(a <= b) { c = (a + ((b - a) / 2)); if((c * c) < num) { a = (c + 1); } else { b = (c - 1); } } ans = a;
false
true
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null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 7, "rule": [ "Rule 2", "Rule 2", "Rule 2", "Rule 6", "Rule 1", "Rule 8", "Rule 5", "Rule 4" ], "state": { "ans": 0, "num": 400, "a": 19, "b": 19, "c": 19, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 8, "rule": [ "Rule 2", "Rule 2", "Rule 7", "Rule 2", "Rule 10", "Rule 33" ], "state": { "ans": 0, "num": 400, "a": 19, "b": 19, "c": 19, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 10, "rule": [ "Rule 2", "Rule 1", "Rule 5", "Rule 4" ], "state": { "ans": 0, "num": 400, "a": 20, "b": 19, "c": 19, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 2", "Rule 13", "Rule 35" ], "state": { "ans": 0, "num": 400, "a": 20, "b": 19, "c": 19, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 17, "rule": [ "Rule 2", "Rule 4" ], "state": { "ans": 20, "num": 400, "a": 20, "b": 19, "c": 19, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 20, "num": 400, "a": 20, "b": 19, "c": 19, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
20
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <num>0</num> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <num>400</num> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>0</b> <c>0</c> <num>400</num> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>400</b> <c>0</c> <num>400</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>400</b> <c>0</c> <num>400</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>400</b> <c>200</c> <num>400</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>400</b> <c>200</c> <num>400</num> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>199</b> <c>200</c> <num>400</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>199</b> <c>200</c> <num>400</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>199</b> <c>99</c> <num>400</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>199</b> <c>99</c> <num>400</num> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>98</b> <c>99</c> <num>400</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>98</b> <c>99</c> <num>400</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>98</b> <c>49</c> <num>400</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>98</b> <c>49</c> <num>400</num> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>48</b> <c>49</c> <num>400</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>48</b> <c>49</c> <num>400</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>48</b> <c>24</c> <num>400</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>48</b> <c>24</c> <num>400</num> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>23</b> <c>24</c> <num>400</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>23</b> <c>24</c> <num>400</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>23</b> <c>11</c> <num>400</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>0</a> <ans>0</ans> <b>23</b> <c>11</c> <num>400</num> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>12</a> <ans>0</ans> <b>23</b> <c>11</c> <num>400</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>12</a> <ans>0</ans> <b>23</b> <c>11</c> <num>400</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>12</a> <ans>0</ans> <b>23</b> <c>17</c> <num>400</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>12</a> <ans>0</ans> <b>23</b> <c>17</c> <num>400</num> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>18</a> <ans>0</ans> <b>23</b> <c>17</c> <num>400</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>18</a> <ans>0</ans> <b>23</b> <c>17</c> <num>400</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>18</a> <ans>0</ans> <b>23</b> <c>20</c> <num>400</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>18</a> <ans>0</ans> <b>23</b> <c>20</c> <num>400</num> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <a>18</a> <ans>0</ans> <b>19</b> <c>20</c> <num>400</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>18</a> <ans>0</ans> <b>19</b> <c>20</c> <num>400</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>18</a> <ans>0</ans> <b>19</b> <c>18</c> <num>400</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>18</a> <ans>0</ans> <b>19</b> <c>18</c> <num>400</num> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>19</a> <ans>0</ans> <b>19</b> <c>18</c> <num>400</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>19</a> <ans>0</ans> <b>19</b> <c>18</c> <num>400</num> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>19</a> <ans>0</ans> <b>19</b> <c>19</c> <num>400</num> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <a>19</a> <ans>0</ans> <b>19</b> <c>19</c> <num>400</num> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <a>20</a> <ans>0</ans> <b>19</b> <c>19</c> <num>400</num> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>20</a> <ans>0</ans> <b>19</b> <c>19</c> <num>400</num> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <a>20</a> <ans>20</ans> <b>19</b> <c>19</c> <num>400</num> </program_state> </step> </ans>
etp-uk-SOS-mbpp_52.imp
int xyz, p, ans, q, r; xyz = 0; p = 0; q = 15; r = 20; while(p < r) { xyz = (xyz + q); p = (p + 1); } ans = xyz;
mbpp_52.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int xyz, p, ans, q, r; xyz = 0; p = 0; q = 15; r = 20; while(p < r) { xyz = (xyz + q); p = (p + 1); } ans = xyz;
false
true
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300
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <p>0</p> <q>0</q> <r>0</r> <xyz>0</xyz> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <p>0</p> <q>0</q> <r>0</r> <xyz>0</xyz> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <p>0</p> <q>0</q> <r>0</r> <xyz>0</xyz> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <p>0</p> <q>15</q> <r>0</r> <xyz>0</xyz> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <p>0</p> <q>15</q> <r>20</r> <xyz>0</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>0</p> <q>15</q> <r>20</r> <xyz>0</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>0</p> <q>15</q> <r>20</r> <xyz>15</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>1</p> <q>15</q> <r>20</r> <xyz>15</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>1</p> <q>15</q> <r>20</r> <xyz>15</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>1</p> <q>15</q> <r>20</r> <xyz>30</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>2</p> <q>15</q> <r>20</r> <xyz>30</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>2</p> <q>15</q> <r>20</r> <xyz>30</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>2</p> <q>15</q> <r>20</r> <xyz>45</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>3</p> <q>15</q> <r>20</r> <xyz>45</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>3</p> <q>15</q> <r>20</r> <xyz>45</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>3</p> <q>15</q> <r>20</r> <xyz>60</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>4</p> <q>15</q> <r>20</r> <xyz>60</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>4</p> <q>15</q> <r>20</r> <xyz>60</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>4</p> <q>15</q> <r>20</r> <xyz>75</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>5</p> <q>15</q> <r>20</r> <xyz>75</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>5</p> <q>15</q> <r>20</r> <xyz>75</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>5</p> <q>15</q> <r>20</r> <xyz>90</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>6</p> <q>15</q> <r>20</r> <xyz>90</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>6</p> <q>15</q> <r>20</r> <xyz>90</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>6</p> <q>15</q> <r>20</r> <xyz>105</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>7</p> <q>15</q> <r>20</r> <xyz>105</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>7</p> <q>15</q> <r>20</r> <xyz>105</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>7</p> <q>15</q> <r>20</r> <xyz>120</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>8</p> <q>15</q> <r>20</r> <xyz>120</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>8</p> <q>15</q> <r>20</r> <xyz>120</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>8</p> <q>15</q> <r>20</r> <xyz>135</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>9</p> <q>15</q> <r>20</r> <xyz>135</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>9</p> <q>15</q> <r>20</r> <xyz>135</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>9</p> <q>15</q> <r>20</r> <xyz>150</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>10</p> <q>15</q> <r>20</r> <xyz>150</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>10</p> <q>15</q> <r>20</r> <xyz>150</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>10</p> <q>15</q> <r>20</r> <xyz>165</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>11</p> <q>15</q> <r>20</r> <xyz>165</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>11</p> <q>15</q> <r>20</r> <xyz>165</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>11</p> <q>15</q> <r>20</r> <xyz>180</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>12</p> <q>15</q> <r>20</r> <xyz>180</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>12</p> <q>15</q> <r>20</r> <xyz>180</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>12</p> <q>15</q> <r>20</r> <xyz>195</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>13</p> <q>15</q> <r>20</r> <xyz>195</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>13</p> <q>15</q> <r>20</r> <xyz>195</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>13</p> <q>15</q> <r>20</r> <xyz>210</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>14</p> <q>15</q> <r>20</r> <xyz>210</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>14</p> <q>15</q> <r>20</r> <xyz>210</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>14</p> <q>15</q> <r>20</r> <xyz>225</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>15</p> <q>15</q> <r>20</r> <xyz>225</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>15</p> <q>15</q> <r>20</r> <xyz>225</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>15</p> <q>15</q> <r>20</r> <xyz>240</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>16</p> <q>15</q> <r>20</r> <xyz>240</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>16</p> <q>15</q> <r>20</r> <xyz>240</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>16</p> <q>15</q> <r>20</r> <xyz>255</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>17</p> <q>15</q> <r>20</r> <xyz>255</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>17</p> <q>15</q> <r>20</r> <xyz>255</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>17</p> <q>15</q> <r>20</r> <xyz>270</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>18</p> <q>15</q> <r>20</r> <xyz>270</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>18</p> <q>15</q> <r>20</r> <xyz>270</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>18</p> <q>15</q> <r>20</r> <xyz>285</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>19</p> <q>15</q> <r>20</r> <xyz>285</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>19</p> <q>15</q> <r>20</r> <xyz>285</xyz> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <p>19</p> <q>15</q> <r>20</r> <xyz>300</xyz> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <p>20</p> <q>15</q> <r>20</r> <xyz>300</xyz> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <p>20</p> <q>15</q> <r>20</r> <xyz>300</xyz> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <ans>300</ans> <p>20</p> <q>15</q> <r>20</r> <xyz>300</xyz> </program_state> </step> </ans>
etp-uk-SOS-mbpp_521.imp
int x, y, z, ans; x = 6; y = 15; z = 20; if(((z % y) != 0) && ((x % z) != 0)) { ans = 1; } else { ans = 0; }
mbpp_521.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int x, y, z, ans; x = 6; y = 15; z = 20; if(((z % y) != 0) && ((x % z) != 0)) { ans = 1; } else { ans = 0; }
false
true
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1
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <x>0</x> <y>0</y> <z>0</z> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <x>6</x> <y>0</y> <z>0</z> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <x>6</x> <y>15</y> <z>0</z> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <x>6</x> <y>15</y> <z>20</z> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <x>6</x> <y>15</y> <z>20</z> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>1</ans> <x>6</x> <y>15</y> <z>20</z> </program_state> </step> </ans>
etp-uk-SOS-mbpp_547.imp
int ans, n; n = 5; while(n > 0) { ans = (ans + n); n = (n / 2); }
mbpp_547.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n; n = 5; while(n > 0) { ans = (ans + n); n = (n / 2); }
false
true
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null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 8", "Rule 4" ], "state": { "ans": 7, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 1", "Rule 14", "Rule 34" ], "state": { "ans": 7, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 5, "rule": [ "Rule 2", "Rule 2", "Rule 5", "Rule 4" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 1, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 6, "rule": [ "Rule 2", "Rule 1", "Rule 8", "Rule 4" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 2", "Rule 1", "Rule 15", "Rule 35" ], "state": { "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 8, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
8
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n>5</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <n>5</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>5</ans> <n>5</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>5</ans> <n>2</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>5</ans> <n>2</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>7</ans> <n>2</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>7</ans> <n>1</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>7</ans> <n>1</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>8</ans> <n>1</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>8</ans> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>8</ans> <n>0</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_55.imp
int x, b, ans, c, d, e; b = 1; c = 2; d = 6; e = 3; x = c; while(b < d) { x = (x * e); if(x < 0) { x = (x + 2); } b = (b + 1); } ans = x;
mbpp_55.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int x, b, ans, c, d, e; b = 1; c = 2; d = 6; e = 3; x = c; while(b < d) { x = (x * e); if(x < 0) { x = (x + 2); } b = (b + 1); } ans = x;
false
true
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{ "ans": 486, "num": null, "a": null, "b": 6, "c": 2, "d": 6, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": 486, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": 3, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
486
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <b>0</b> <c>0</c> <d>0</d> <e>0</e> <x>0</x> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <b>1</b> <c>0</c> <d>0</d> <e>0</e> <x>0</x> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <b>1</b> <c>2</c> <d>0</d> <e>0</e> <x>0</x> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <b>1</b> <c>2</c> <d>6</d> <e>0</e> <x>0</x> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <b>1</b> <c>2</c> <d>6</d> <e>3</e> <x>0</x> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <b>1</b> <c>2</c> <d>6</d> <e>3</e> <x>2</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <b>1</b> <c>2</c> <d>6</d> <e>3</e> <x>2</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <b>1</b> <c>2</c> <d>6</d> <e>3</e> <x>6</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <b>1</b> <c>2</c> <d>6</d> <e>3</e> <x>6</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <b>2</b> <c>2</c> <d>6</d> <e>3</e> <x>6</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <b>2</b> <c>2</c> <d>6</d> <e>3</e> <x>6</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <b>2</b> <c>2</c> <d>6</d> <e>3</e> <x>18</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <b>2</b> <c>2</c> <d>6</d> <e>3</e> <x>18</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <b>3</b> <c>2</c> <d>6</d> <e>3</e> <x>18</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <b>3</b> <c>2</c> <d>6</d> <e>3</e> <x>18</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <b>3</b> <c>2</c> <d>6</d> <e>3</e> <x>54</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <b>3</b> <c>2</c> <d>6</d> <e>3</e> <x>54</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <b>4</b> <c>2</c> <d>6</d> <e>3</e> <x>54</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <b>4</b> <c>2</c> <d>6</d> <e>3</e> <x>54</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <b>4</b> <c>2</c> <d>6</d> <e>3</e> <x>162</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <b>4</b> <c>2</c> <d>6</d> <e>3</e> <x>162</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <b>5</b> <c>2</c> <d>6</d> <e>3</e> <x>162</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <b>5</b> <c>2</c> <d>6</d> <e>3</e> <x>162</x> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <b>5</b> <c>2</c> <d>6</d> <e>3</e> <x>486</x> </program_state> </step> <step> <linenumber>10</linenumber> <program_state> <ans>0</ans> <b>5</b> <c>2</c> <d>6</d> <e>3</e> <x>486</x> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <b>6</b> <c>2</c> <d>6</d> <e>3</e> <x>486</x> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <b>6</b> <c>2</c> <d>6</d> <e>3</e> <x>486</x> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <ans>486</ans> <b>6</b> <c>2</c> <d>6</d> <e>3</e> <x>486</x> </program_state> </step> </ans>
etp-uk-SOS-mbpp_558.imp
int ans, n1, n2; n1 = 123; n2 = 256; while(n1 > 0) { ans = (ans + ((n2 % 10) - (n1 % 10))); n1 = (n1 / 10); n2 = (n2 / 10); }
mbpp_558.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n1, n2; n1 = 123; n2 = 256; while(n1 > 0) { ans = (ans + ((n2 % 10) - (n1 % 10))); n1 = (n1 / 10); n2 = (n2 / 10); }
false
true
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"temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 7, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": 0, "n2": 0, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
7
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n1>0</n1> <n2>0</n2> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n1>123</n1> <n2>0</n2> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <n1>123</n1> <n2>256</n2> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <n1>123</n1> <n2>256</n2> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>3</ans> <n1>123</n1> <n2>256</n2> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>3</ans> <n1>12</n1> <n2>256</n2> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>3</ans> <n1>12</n1> <n2>25</n2> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>3</ans> <n1>12</n1> <n2>25</n2> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>6</ans> <n1>12</n1> <n2>25</n2> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>6</ans> <n1>1</n1> <n2>25</n2> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>6</ans> <n1>1</n1> <n2>2</n2> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>6</ans> <n1>1</n1> <n2>2</n2> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>7</ans> <n1>1</n1> <n2>2</n2> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>7</ans> <n1>0</n1> <n2>2</n2> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>7</ans> <n1>0</n1> <n2>0</n2> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>7</ans> <n1>0</n1> <n2>0</n2> </program_state> </step> </ans>
etp-uk-SOS-mbpp_566.imp
int ans, n; n = 97; while(n > 0) { ans = (ans + (n % 10)); n = (n / 10); }
mbpp_566.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n; n = 97; while(n > 0) { ans = (ans + (n % 10)); n = (n / 10); }
false
true
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16
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n>97</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <n>97</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>7</ans> <n>97</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>7</ans> <n>9</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>7</ans> <n>9</n> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>16</ans> <n>9</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>16</ans> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>16</ans> <n>0</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_575.imp
int ans, n, a, c, i, l, r; a = 3; n = 6; l = 4; r = 20; i = l; while(i <= r) { if((i % a) != 0) { c = (c + 1); } if(c == n) { ans = i; halt; } i = (i + 1); } ans = (- 1);
mbpp_575.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n, a, c, i, l, r; a = 3; n = 6; l = 4; r = 20; i = l; while(i <= r) { if((i % a) != 0) { c = (c + 1); } if(c == n) { ans = i; halt; } i = (i + 1); } ans = (- 1);
false
true
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null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 16, "rule": [ "Rule 41" ], "state": { "ans": 11, "num": null, "a": 3, "b": null, "c": 6, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": 20, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": 11, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": 4, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 11, "num": null, "a": 3, "b": null, "c": 6, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": 20, "x": null, "B": null, "X": null, "n": 6, "y": null, "xyz": null, "i": 11, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": 4, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
11
<ans> <step> <linenumber>1</linenumber> <program_state> <a>0</a> <ans>0</ans> <c>0</c> <i>0</i> <l>0</l> <n>0</n> <r>0</r> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>0</c> <i>0</i> <l>0</l> <n>0</n> <r>0</r> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>0</c> <i>0</i> <l>0</l> <n>6</n> <r>0</r> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>0</c> <i>0</i> <l>4</l> <n>6</n> <r>0</r> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>0</c> <i>0</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>0</c> <i>4</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>0</c> <i>4</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>0</c> <i>4</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>1</c> <i>4</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>1</c> <i>4</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>1</c> <i>5</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>1</c> <i>5</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>1</c> <i>5</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>2</c> <i>5</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>2</c> <i>5</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>2</c> <i>6</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>2</c> <i>6</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>2</c> <i>6</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>2</c> <i>6</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>2</c> <i>7</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>2</c> <i>7</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>2</c> <i>7</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>3</c> <i>7</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>3</c> <i>7</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>3</c> <i>8</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>3</c> <i>8</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>3</c> <i>8</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>4</c> <i>8</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>4</c> <i>8</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>4</c> <i>9</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>4</c> <i>9</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>4</c> <i>9</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>4</c> <i>9</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>4</c> <i>10</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>4</c> <i>10</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>4</c> <i>10</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>5</c> <i>10</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>5</c> <i>10</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>18</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>5</c> <i>11</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>5</c> <i>11</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>5</c> <i>11</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>11</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>6</c> <i>11</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <a>3</a> <ans>0</ans> <c>6</c> <i>11</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>15</linenumber> <program_state> <a>3</a> <ans>11</ans> <c>6</c> <i>11</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> <step> <linenumber>16</linenumber> <program_state> <a>3</a> <ans>11</ans> <c>6</c> <i>11</i> <l>4</l> <n>6</n> <r>20</r> </program_state> </step> </ans>
etp-uk-SOS-mbpp_59.imp
int ans, n; n = 10; ans = (((3 * n) * n) - (2 * n));
mbpp_59.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n; n = 10; ans = (((3 * n) * n) - (2 * n));
false
true
[ { "line_number": 1, "rule": [ "Rule 38", "Rule 3", "Rule 3" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 2, "rule": [ "Rule 1", "Rule 4" ], "state": { "ans": 0, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } }, { "line_number": 3, "rule": [ "Rule 1", "Rule 2", "Rule 7", "Rule 2", "Rule 7", "Rule 1", "Rule 2", "Rule 7", "Rule 6", "Rule 4" ], "state": { "ans": 280, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null } } ]
{ "ans": 280, "num": null, "a": null, "b": null, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 10, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
280
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <n>10</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>280</ans> <n>10</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_600.imp
int ans, n, b; b = 1; n = 2; while(n > 0) { if((n % 2) == 0) { b = (b * 2); n = (n / 2); } else { b = (b * 3); n = (n / 3); } } if((b % 2) == 0) { ans = 1; }
mbpp_600.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int ans, n, b; b = 1; n = 2; while(n > 0) { if((n % 2) == 0) { b = (b * 2); n = (n / 2); } else { b = (b * 3); n = (n / 3); } } if((b % 2) == 0) { ans = 1; }
false
true
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{ "ans": 1, "num": null, "a": null, "b": 6, "c": null, "d": null, "f": null, "j": null, "k": null, "m": null, "p": null, "q": null, "r": null, "x": null, "B": null, "X": null, "n": 0, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": null, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": null, "r2": null, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
1
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <b>0</b> <n>0</n> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <b>1</b> <n>0</n> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <b>1</b> <n>2</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <b>1</b> <n>2</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <b>1</b> <n>2</n> </program_state> </step> <step> <linenumber>8</linenumber> <program_state> <ans>0</ans> <b>2</b> <n>2</n> </program_state> </step> <step> <linenumber>9</linenumber> <program_state> <ans>0</ans> <b>2</b> <n>1</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <b>2</b> <n>1</n> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <b>2</b> <n>1</n> </program_state> </step> <step> <linenumber>13</linenumber> <program_state> <ans>0</ans> <b>6</b> <n>1</n> </program_state> </step> <step> <linenumber>14</linenumber> <program_state> <ans>0</ans> <b>6</b> <n>0</n> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <b>6</b> <n>0</n> </program_state> </step> <step> <linenumber>17</linenumber> <program_state> <ans>0</ans> <b>6</b> <n>0</n> </program_state> </step> <step> <linenumber>19</linenumber> <program_state> <ans>1</ans> <b>6</b> <n>0</n> </program_state> </step> </ans>
etp-uk-SOS-mbpp_626.imp
int r, l, r2, l2, ans, f; r = 2; l = (r - 1); r2 = (r * 2); l2 = (l * 2); f = 0; while((l2 - l) > 1) { if((l % 2) == 0) { l = (l / 2); } else { l = (((l * 3) - r) + 1); } if(l2 > l) { ans = (- 1); halt; } f = (l * (l + l2)); if(f > r2) { ans = f; halt; } } ans = r2;
mbpp_626.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int r, l, r2, l2, ans, f; r = 2; l = (r - 1); r2 = (r * 2); l2 = (l * 2); f = 0; while((l2 - l) > 1) { if((l % 2) == 0) { l = (l / 2); } else { l = (((l * 3) - r) + 1); } if(l2 > l) { ans = (- 1); halt; } f = (l * (l + l2)); if(f > r2) { ans = f; halt; } } ans = r2;
false
true
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{ "ans": 4, "num": null, "a": null, "b": null, "c": null, "d": null, "f": 0, "j": null, "k": null, "m": null, "p": null, "q": null, "r": 2, "x": null, "B": null, "X": null, "n": null, "y": null, "xyz": null, "i": null, "s": null, "t": null, "first": null, "tk": null, "T": null, "e": null, "h": null, "l": 1, "count": null, "sum": null, "k9m": null, "p2d": null, "x7f": null, "aa": null, "bb": null, "tmp": null, "x1": null, "x2": null, "y1": null, "y2": null, "cnt": null, "vol": null, "z": null, "n1": null, "n2": null, "l2": 2, "r2": 4, "temp": null, "n3": null, "number": null, "ax1": null, "ax2": null, "ay1": null, "ay2": null, "bx1": null, "bx2": null, "by1": null, "by2": null, "left": null, "oh": null, "ow": null, "right": null, "r1x1": null, "r1x2": null, "r1y1": null, "r1y2": null, "r2x1": null, "r2x2": null, "r2y1": null, "r2y2": null }
4
<ans> <step> <linenumber>1</linenumber> <program_state> <ans>0</ans> <f>0</f> <l2>0</l2> <l>0</l> <r2>0</r2> <r>0</r> </program_state> </step> <step> <linenumber>2</linenumber> <program_state> <ans>0</ans> <f>0</f> <l2>0</l2> <l>0</l> <r2>0</r2> <r>2</r> </program_state> </step> <step> <linenumber>3</linenumber> <program_state> <ans>0</ans> <f>0</f> <l2>0</l2> <l>1</l> <r2>0</r2> <r>2</r> </program_state> </step> <step> <linenumber>4</linenumber> <program_state> <ans>0</ans> <f>0</f> <l2>0</l2> <l>1</l> <r2>4</r2> <r>2</r> </program_state> </step> <step> <linenumber>5</linenumber> <program_state> <ans>0</ans> <f>0</f> <l2>2</l2> <l>1</l> <r2>4</r2> <r>2</r> </program_state> </step> <step> <linenumber>6</linenumber> <program_state> <ans>0</ans> <f>0</f> <l2>2</l2> <l>1</l> <r2>4</r2> <r>2</r> </program_state> </step> <step> <linenumber>7</linenumber> <program_state> <ans>0</ans> <f>0</f> <l2>2</l2> <l>1</l> <r2>4</r2> <r>2</r> </program_state> </step> <step> <linenumber>29</linenumber> <program_state> <ans>4</ans> <f>0</f> <l2>2</l2> <l>1</l> <r2>4</r2> <r>2</r> </program_state> </step> </ans>
etp-uk-SOS-mbpp_634.imp
int n, i, j, ans; i = 1; n = 3; while(i <= n) { j = (2 * i); ans = (ans + (((j * j) * j) * j)); i = (i + 1); }
mbpp_634.imp
SOS
<program> ::= <decl> <stmt_list> <decl> ::= 'int' <SP> <ids> <SP> ';' <ids> ::= <id> <id_list> <id_list> ::= <SP> ',' <SP> <id> <id_list> | Ξ΅ <stmt_list> ::= <stmt> <stmt_list> | Ξ΅ <stmt> ::= <block> | <id> <SP> '=' <SP> <aexp> <SP> ';' | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> <SP> 'else' <SP> <block> | 'if' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'while' <SP> '(' <SP> <bexp> <SP> ')' <SP> <block> | 'halt' ';' | 'continue' ';' | 'break' ';' <block> ::= '{' <SP> <stmt_list> <SP> '}' <aexp> ::= <id> | <literal> | '(' <SP> <aexp> <SP> <mathop> <SP> <aexp> <SP> ')' | '+' <aexp> | '-' <aexp> <bexp> ::= <bool> | '(' <SP> <aexp> <SP> <relop> <SP> <aexp> <SP> ')' | '(' <SP> <lognot> <SP> <bexp> <SP> ')' | '(' <SP> <bexp> <SP> <logicalop> <SP> <bexp> <SP> ')' <bool> ::= 'true' | 'false' <mathop> ::= '+' | '-' | '*' | '/' | '%' <relop> ::= '<' | '<=' | '>' | '>=' | '==' | '!=' <lognot> ::= '!' <logicalop> ::= '&&' | '||' <SP> ::= ' ' | '\\t' <id> ::= <letter> <alphanum> <alphanum> ::= <letter> <alphanum> | <digit> <alphanum> | Ξ΅ <literal> ::= <digit> <digits> <digits> ::= <digit> <digits> | Ξ΅ <letter> ::= 'a' | 'b' | 'c' | 'd' | 'e' | 'f' | 'g' | 'h' | 'i' | 'j' | 'k' | 'l' | 'm' | 'n' | 'o' | 'p' | 'q' | 'r' | 's' | 't' | 'u' | 'v' | 'w' | 'x' | 'y' | 'z' | 'A' | 'B' | 'C' | 'D' | 'E' | 'F' | 'G' | 'H' | 'I' | 'J' | 'K' | 'L' | 'M' | 'N' | 'O' | 'P' | 'Q' | 'R' | 'S' | 'T' | 'U' | 'V' | 'W' | 'X' | 'Y' | 'Z' <digit> ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
β„€ := Set of integers 𝔹 := {true, false} , set of booleans 𝜎 := { ID ↦ β„€ } v := integer value x := integer variable e := integer expression q := boolean value b := boolean expression S := statement configuration := 〈operation,𝜎βŒͺ single-step transition := 〈e,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ, e evaluates to v in exactly one step multi-step transition := 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ, e evaluates to v through multiple single-step transitions empty computation := 〈Ρ,𝜎βŒͺ halt execution := 〈HALT,𝜎βŒͺ, halt program execution Rule 1 := { v ∈ β„€ -------------- 〈v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 2 := { 〈x,𝜎βŒͺ 𝜎(x) = v v ∈ β„€ ------------------------- 〈x,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 3 := { 𝜎(x) = βŸ‚ --------------------------- 〈int x;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ 0]βŒͺ } Rule 4 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) β‰  βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈Ρ,𝜎[x ↦ v]βŒͺ } Rule 5 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 + e2,𝜎βŒͺ β†’ 〈v1 + e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 + e2,𝜎βŒͺ β†’ 〈v1 + v2,𝜎βŒͺ v3 = v1 + v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 + v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 6 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 - e2,𝜎βŒͺ β†’ 〈v1 - e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 - e2,𝜎βŒͺ β†’ 〈v1 - v2,𝜎βŒͺ v3 = v1 - v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 - v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 7 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------- 〈e1 * e2,𝜎βŒͺ β†’ 〈v1 * e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------- 〈v1 * e2,𝜎βŒͺ β†’ 〈v1 * v2,𝜎βŒͺ v3 = v1 * v2 v3 ∈ β„€ ---------------------------------------------------- 〈v1 * v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 8 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ v3 = v1 / v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 9 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ v3 = v1 % v2 v3 ∈ β„€ β‹€ v2 β‰  0 ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈v3,𝜎βŒͺ } Rule 10 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ----------------------------------------------------------------- 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 < v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 11 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 < e2,𝜎βŒͺ β†’ 〈v1 < e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < e2,𝜎βŒͺ β†’ 〈v1 < v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 < v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 12 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 13 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ------------------------------------------------------------------ 〈e1 <= e2,𝜎βŒͺ β†’ 〈v1 <= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= e2,𝜎βŒͺ β†’ 〈v1 <= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 <= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 14 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 > v2 ----------------------------------------------------------------- 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ----------------------------------------------------------------- 〈v1 > v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 15 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 <= v2 ------------------------------------------------------------------ 〈e1 > e2,𝜎βŒͺ β†’ 〈v1 > e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > e2,𝜎βŒͺ β†’ 〈v1 > v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 > v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 16 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 >= v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 17 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 < v2 ------------------------------------------------------------------ 〈e1 >= e2,𝜎βŒͺ β†’ 〈v1 >= e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= e2,𝜎βŒͺ β†’ 〈v1 >= v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 >= v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 18 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 19 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 == e2,𝜎βŒͺ β†’ 〈v1 == e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == e2,𝜎βŒͺ β†’ 〈v1 == v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 == v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 20 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 β‰  v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 21 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v1 = v2 ------------------------------------------------------------------ 〈e1 != e2,𝜎βŒͺ β†’ 〈v1 != e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != e2,𝜎βŒͺ β†’ 〈v1 != v2,𝜎βŒͺ ------------------------------------------------------------------ 〈v1 != v2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 22 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true β‹€ q2 = true --------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 23 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false ⋁ q2 = false ----------------------------------------------------------------------------------- 〈b1 && b2,𝜎βŒͺ β†’ 〈q1 && b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && b2,𝜎βŒͺ β†’ 〈q1 && q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 && q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 24 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = true ⋁ q2 = true --------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ --------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 25 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q1,q2 ∈ 𝔹 q1 = false β‹€ q2 = false ----------------------------------------------------------------------------------- 〈b1 || b2,𝜎βŒͺ β†’ 〈q1 || b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || b2,𝜎βŒͺ β†’ 〈q1 || q2,𝜎βŒͺ ----------------------------------------------------------------------------------- 〈q1 || q2,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 26 := { ---------------------- 〈true,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 27 := { ---------------------- 〈false,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 28 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈false,𝜎βŒͺ } Rule 29 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈!b,𝜎βŒͺ β†’ 〈true,𝜎βŒͺ } Rule 30 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 31 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------- 〈if(b){S},𝜎βŒͺ β†’ 〈S,𝜎βŒͺ } Rule 32 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false -------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S2,𝜎βŒͺ } Rule 33 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------- 〈if(b){S1} else{S2},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 34 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true ------------------------------------------ 〈while(b){S1},𝜎βŒͺ β†’ 〈S1;while(b){S1},𝜎βŒͺ } Rule 35 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = false ------------------------------------------- 〈while(b){S1},𝜎βŒͺ β†’ 〈Ρ,𝜎βŒͺ } Rule 36 := { 〈S1,𝜎βŒͺ β­‡ 〈Ρ,𝜎1βŒͺ 〈S2,𝜎1βŒͺ β­‡ 〈Ρ,𝜎2βŒͺ ----------------------------------------- 〈S1;S2,𝜎βŒͺ β†’ 〈S2,𝜎1βŒͺ ----------------------------------------- 〈S2,𝜎1βŒͺ β†’ 〈Ρ,𝜎2βŒͺ } Rule 37 := { --------------------- 〈{S1},𝜎βŒͺ β†’ 〈S1,𝜎βŒͺ } Rule 38 := { 𝜎(x1) = βŸ‚ β‹€ 𝜎(x2) = βŸ‚ -------------------------------------- 〈int x1,x2;,𝜎βŒͺ β†’ 〈int x1; int x2;,𝜎βŒͺ } Rule 39 := { 𝜎(x) = βŸ‚ ------------------------ 〈x,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 40 := { 〈e,𝜎βŒͺ β­‡ 〈v,𝜎βŒͺ v ∈ β„€ 𝜎(x) = βŸ‚ ------------------------------------- 〈x=e;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 41 := { ---------------------- 〈halt;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 42 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true --------------------------------------------------------------------- 〈while(b){S1;continue;S2},𝜎βŒͺ β†’ 〈S1;while(b){S1;continue;S2},𝜎βŒͺ } Rule 43 := { 〈b,𝜎βŒͺ β­‡ 〈q,𝜎βŒͺ q ∈ 𝔹 q = true -------------------------------------------- 〈while(b){S1;break;S2},𝜎βŒͺ β†’ 〈S1;,𝜎βŒͺ } Rule 44 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 45 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο΄ ∈ {/, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ e2 Ο΄ e3,𝜎βŒͺ β†’ 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο΄ v2 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ e3,𝜎βŒͺ β†’ 〈v4 Ο΄ v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο΄ v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 46 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† e2 Ο† e3,𝜎βŒͺ β†’ 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ v4 = v1 Ο΄ v2 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v2 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† e3,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† e3,𝜎βŒͺ β†’ 〈v4 Ο† v3,𝜎βŒͺ v5 = v4 Ο΄ v3 -------------------------------------------------------------------------------------------------------------- 〈v4 Ο† v3,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 47 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ Ο† ∈ {+, -, /, *, %} v1,v2,v3,v4,v5 ∈ β„€ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (e2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e3,𝜎βŒͺ β­‡ 〈v3,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† e3),𝜎βŒͺ β†’ 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ v4 = v2 Ο† v3 -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† (v2 Ο† v3),𝜎βŒͺ β†’ 〈e1 Ο† v4,𝜎βŒͺ 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ -------------------------------------------------------------------------------------------------------------- 〈e1 Ο† v4,𝜎βŒͺ β†’ 〈v1 Ο† v4,𝜎βŒͺ v5 = v1 Ο† v4 -------------------------------------------------------------------------------------------------------------- 〈v1 Ο† v4,𝜎βŒͺ β†’ 〈v5,𝜎βŒͺ } Rule 48 := { v1,v2 ∈ β„€ v2 = -v1 ----------------------- 〈-v1,𝜎βŒͺ β†’ 〈v2,𝜎βŒͺ } Rule 49 := { v ∈ β„€ ---------------- 〈+v,𝜎βŒͺ β†’ 〈v,𝜎βŒͺ } Rule 50 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ Ο΄ ∈ {&&, ||} q1,q2,q3,q4 ∈ 𝔹 ------------------------------------------------------------------------------------------- 〈! b1 Ο΄ b2,𝜎βŒͺ β†’ 〈! q1 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ q3 = ! q1 ------------------------------------------------------------------------------------------- 〈! q1 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ b2,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ ------------------------------------------------------------------------------------------- 〈q3 Ο΄ b2,𝜎βŒͺ β†’ 〈q3 Ο΄ q2,𝜎βŒͺ q4 = q3 Ο΄ q2 ------------------------------------------------------------------------------------------- 〈q3 Ο΄ q2,𝜎βŒͺβ†’ 〈q4,𝜎βŒͺ } Rule 51 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------ 〈b1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† b2 Ο† b3,𝜎βŒͺ β†’ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ q4 = q1 Ο΄ q2 ------------------------------------------------------------------------------------------------------------------ 〈q1 Ο† q2 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† b3,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† b3,𝜎βŒͺ β†’ 〈q4 Ο† q3,𝜎βŒͺ q5 = q4 Ο΄ q3 ------------------------------------------------------------------------------------------------------------------ 〈q4 Ο† q3,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 52 := { 〈b1,𝜎βŒͺ β­‡ 〈q1,𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ Ο† ∈ {&&, ||} q1,q2,q3,q4,q5 ∈ 𝔹 ------------------------------------------------------------------------------------------------------------------- 〈b1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ 〈b2,𝜎βŒͺ β­‡ 〈q2,𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (b2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ 〈b3,𝜎βŒͺ β­‡ 〈q3,𝜎βŒͺ ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† b3),𝜎βŒͺ β†’ 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ q4 = q2 Ο† q3 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† (q2 Ο† q3),𝜎βŒͺ β†’ 〈q1 Ο† q4,𝜎βŒͺ q5 = q1 Ο† q4 ------------------------------------------------------------------------------------------------------------------- 〈q1 Ο† q4,𝜎βŒͺ β†’ 〈q5,𝜎βŒͺ } Rule 53 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 / e2,𝜎βŒͺ β†’ 〈v1 / e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / e2,𝜎βŒͺ β†’ 〈v1 / v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 / v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 54 := { 〈e1,𝜎βŒͺ β­‡ 〈v1,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ v1,v2 ∈ β„€ β‹€ v2 = 0 ---------------------------------------------------------------- 〈e1 % e2,𝜎βŒͺ β†’ 〈v1 % e2,𝜎βŒͺ 〈e2,𝜎βŒͺ β­‡ 〈v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % e2,𝜎βŒͺ β†’ 〈v1 % v2,𝜎βŒͺ ---------------------------------------------------------------- 〈v1 % v2,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 55 := { ----------------------------- 〈continue;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ } Rule 56 := { -------------------------- 〈break;,𝜎βŒͺ β†’ 〈HALT,𝜎βŒͺ }
IMP_SOS
int n, i, j, ans; i = 1; n = 3; while(i <= n) { j = (2 * i); ans = (ans + (((j * j) * j) * j)); i = (i + 1); }
false
true
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