Datasets:

internlm
/

Lean-Github

Dataset card Data Studio Files Files and versions Community

Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro H' a2	case a Δ : List Formula F : Formula Δ_1 Δ_2 : List Formula H_1 H_2 phi✝ : Formula a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H ⊢ ∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H	case a Δ : List Formula F : Formula Δ_1 Δ_2 : List Formula H_1 H_2 phi✝ : Formula a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H H' : Formula a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2 ⊢ Holds D I V E H'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp at a1	case a Δ : List Formula F : Formula Δ_1 Δ_2 : List Formula H_1 H_2 phi✝ : Formula a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H H' : Formula a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2 ⊢ Holds D I V E H'	case a Δ : List Formula F : Formula Δ_1 Δ_2 : List Formula H_1 H_2 phi✝ : Formula a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env H' : Formula a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2 a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H ⊢ Holds D I V E H'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	apply a1	case a Δ : List Formula F : Formula Δ_1 Δ_2 : List Formula H_1 H_2 phi✝ : Formula a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env H' : Formula a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2 a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H ⊢ Holds D I V E H'	case a.a Δ : List Formula F : Formula Δ_1 Δ_2 : List Formula H_1 H_2 phi✝ : Formula a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env H' : Formula a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2 a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H ⊢ H' ∈ Δ_1 ∨ H' = H_2 ∨ H' = H_1 ∨ H' ∈ Δ_2
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp at a2	case a.a Δ : List Formula F : Formula Δ_1 Δ_2 : List Formula H_1 H_2 phi✝ : Formula a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env H' : Formula a2 : H' ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2 a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H ⊢ H' ∈ Δ_1 ∨ H' = H_2 ∨ H' = H_1 ∨ H' ∈ Δ_2	case a.a Δ : List Formula F : Formula Δ_1 Δ_2 : List Formula H_1 H_2 phi✝ : Formula a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env H' : Formula a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H a2 : H' ∈ Δ_1 ∨ H' = H_1 ∨ H' = H_2 ∨ H' ∈ Δ_2 ⊢ H' ∈ Δ_1 ∨ H' = H_2 ∨ H' = H_1 ∨ H' ∈ Δ_2
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	tauto	case a.a Δ : List Formula F : Formula Δ_1 Δ_2 : List Formula H_1 H_2 phi✝ : Formula a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env H' : Formula a1 : ∀ (H : Formula), H ∈ Δ_1 ∨ H = H_2 ∨ H = H_1 ∨ H ∈ Δ_2 → Holds D I V E H a2 : H' ∈ Δ_1 ∨ H' = H_1 ∨ H' = H_2 ∨ H' ∈ Δ_2 ⊢ H' ∈ Δ_1 ∨ H' = H_2 ∨ H' = H_1 ∨ H' ∈ Δ_2	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E a1	Δ : List Formula F phi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [phi], Holds D I V E H) → Holds D I V E phi	Δ : List Formula F phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ [phi], Holds D I V E H ⊢ Holds D I V E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp at a1	Δ : List Formula F phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ [phi], Holds D I V E H ⊢ Holds D I V E phi	Δ : List Formula F phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a1 : Holds D I V E phi ⊢ Holds D I V E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	exact a1	Δ : List Formula F phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a1 : Holds D I V E phi ⊢ Holds D I V E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E true_	Δ : List Formula F : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E true_
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E true_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F phi psi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (phi.imp_ (psi.imp_ phi))	Δ : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (phi.imp_ (psi.imp_ phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (phi.imp_ (psi.imp_ phi))	Δ : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E phi → Holds D I V E psi → Holds D I V E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	tauto	Δ : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E phi → Holds D I V E psi → Holds D I V E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F phi psi chi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi)))	Δ : List Formula F phi psi chi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E ((phi.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi)))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F phi psi chi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E ((phi.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi)))	Δ : List Formula F phi psi chi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ (Holds D I V E phi → Holds D I V E psi → Holds D I V E chi) → (Holds D I V E phi → Holds D I V E psi) → Holds D I V E phi → Holds D I V E chi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	tauto	Δ : List Formula F phi psi chi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ (Holds D I V E phi → Holds D I V E psi → Holds D I V E chi) → (Holds D I V E phi → Holds D I V E psi) → Holds D I V E phi → Holds D I V E chi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F phi psi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi.not_.imp_ psi.not_).imp_ (psi.imp_ phi))	Δ : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E ((phi.not_.imp_ psi.not_).imp_ (psi.imp_ phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E ((phi.not_.imp_ psi.not_).imp_ (psi.imp_ phi))	Δ : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ (¬Holds D I V E phi → ¬Holds D I V E psi) → Holds D I V E psi → Holds D I V E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	tauto	Δ : List Formula F phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ (¬Holds D I V E phi → ¬Holds D I V E psi) → Holds D I V E psi → Holds D I V E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E a1	Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E (phi.imp_ psi) ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E psi	Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E (phi.imp_ psi) ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ⊢ Holds D I V E psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds] at ih_major	Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E (phi.imp_ psi) ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ⊢ Holds D I V E psi	Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi ⊢ Holds D I V E psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	apply ih_major	Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi ⊢ Holds D I V E psi	case a Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi ⊢ ∀ H ∈ Δ', Holds D I V E H case a Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi ⊢ Holds D I V E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro H' a2	case a Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi ⊢ ∀ H ∈ Δ', Holds D I V E H	case a Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi H' : Formula a2 : H' ∈ Δ' ⊢ Holds D I V E H'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	exact a1 H' a2	case a Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi H' : Formula a2 : H' ∈ Δ' ⊢ Holds D I V E H'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	apply ih_minor	case a Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi ⊢ Holds D I V E phi	case a.a Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi ⊢ ∀ H ∈ Δ', Holds D I V E H
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	exact a1	case a.a Δ : List Formula F : Formula Δ' : List Formula phi psi : Formula a✝¹ : IsDeduct Δ' (phi.imp_ psi) a✝ : IsDeduct Δ' phi ih_minor : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ih_major : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi → Holds D I V E psi ⊢ ∀ H ∈ Δ', Holds D I V E H	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E a1	Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E (H.imp_ phi)	Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ⊢ Holds D I V E (H.imp_ phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ⊢ Holds D I V E (H.imp_ phi)	Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ⊢ Holds D I V E H → Holds D I V E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro a2	Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H ⊢ Holds D I V E H → Holds D I V E phi	Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H a2 : Holds D I V E H ⊢ Holds D I V E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	apply ih_2	Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H a2 : Holds D I V E H ⊢ Holds D I V E phi	case a Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H a2 : Holds D I V E H ⊢ ∀ H_1 ∈ H :: Δ', Holds D I V E H_1
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp	case a Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H a2 : Holds D I V E H ⊢ ∀ H_1 ∈ H :: Δ', Holds D I V E H_1	case a Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H a2 : Holds D I V E H ⊢ Holds D I V E H ∧ ∀ a ∈ Δ', Holds D I V E a
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	constructor	case a Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H a2 : Holds D I V E H ⊢ Holds D I V E H ∧ ∀ a ∈ Δ', Holds D I V E a	case a.left Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H a2 : Holds D I V E H ⊢ Holds D I V E H case a.right Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H a2 : Holds D I V E H ⊢ ∀ a ∈ Δ', Holds D I V E a
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	exact a2	case a.left Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H a2 : Holds D I V E H ⊢ Holds D I V E H	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	exact a1	case a.right Δ : List Formula F : Formula Δ' : List Formula H phi : Formula a✝ : IsDeduct (H :: Δ') phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H ∈ Δ', Holds D I V E H a2 : Holds D I V E H ⊢ ∀ a ∈ Δ', Holds D I V E a	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F : Formula v : VarName phi psi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi)))	Δ : List Formula F : Formula v : VarName phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi)))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F : Formula v : VarName phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi)))	Δ : List Formula F : Formula v : VarName phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ (∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi) → (∀ (d : D), Holds D I (Function.updateITE V v d) E phi) → ∀ (d : D), Holds D I (Function.updateITE V v d) E psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro a2 a3 d	Δ : List Formula F : Formula v : VarName phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ (∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi) → (∀ (d : D), Holds D I (Function.updateITE V v d) E phi) → ∀ (d : D), Holds D I (Function.updateITE V v d) E psi	Δ : List Formula F : Formula v : VarName phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi a3 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi d : D ⊢ Holds D I (Function.updateITE V v d) E psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	apply a2 d	Δ : List Formula F : Formula v : VarName phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi a3 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi d : D ⊢ Holds D I (Function.updateITE V v d) E psi	Δ : List Formula F : Formula v : VarName phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi a3 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi d : D ⊢ Holds D I (Function.updateITE V v d) E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	exact a3 d	Δ : List Formula F : Formula v : VarName phi psi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi → Holds D I (Function.updateITE V v d) E psi a3 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi d : D ⊢ Holds D I (Function.updateITE V v d) E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F : Formula v t : VarName phi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi))	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	obtain s1 := FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem D I V E (Function.updateITE id v t) freshChar phi	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi))	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi ⊢ Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi ⊢ Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi))	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi ⊢ (∀ (d : D), Holds D I (Function.updateITE V v d) E phi) → Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro a2	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi ⊢ (∀ (d : D), Holds D I (Function.updateITE V v d) E phi) → Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi ⊢ Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [s1]	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi ⊢ Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi ⊢ Holds D I (V ∘ Function.updateITE id v t) E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Function.updateITE_comp_left]	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi ⊢ Holds D I (V ∘ Function.updateITE id v t) E phi	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi ⊢ Holds D I (Function.updateITE (V ∘ id) v (V t)) E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi ⊢ Holds D I (Function.updateITE (V ∘ id) v (V t)) E phi	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi ⊢ Holds D I (Function.updateITE V v (V t)) E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	exact a2 (V t)	Δ : List Formula F : Formula v t : VarName phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H s1 : Holds D I V E (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi) ↔ Holds D I (V ∘ Function.updateITE id v t) E phi a2 : ∀ (d : D), Holds D I (Function.updateITE V v d) E phi ⊢ Holds D I (Function.updateITE V v (V t)) E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (phi.imp_ (forall_ v phi))	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (phi.imp_ (forall_ v phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (phi.imp_ (forall_ v phi))	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E phi → ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro a2 d	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E phi → ∀ (d : D), Holds D I (Function.updateITE V v d) E phi	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D ⊢ Holds D I (Function.updateITE V v d) E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	have s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D ⊢ Holds D I (Function.updateITE V v d) E phi	case s1 Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D ⊢ Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi ⊢ Holds D I (Function.updateITE V v d) E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	{ apply Holds_coincide_Var intro v' a1 simp only [Function.updateITE] split_ifs case _ c1 => subst c1 contradiction case _ c1 => rfl }	case s1 Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D ⊢ Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi ⊢ Holds D I (Function.updateITE V v d) E phi	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi ⊢ Holds D I (Function.updateITE V v d) E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [s1]	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi ⊢ Holds D I (Function.updateITE V v d) E phi	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi ⊢ Holds D I V E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	exact a2	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi ⊢ Holds D I V E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	apply Holds_coincide_Var	case s1 Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D ⊢ Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi	case s1.h1 Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D ⊢ ∀ (v_1 : VarName), isFreeIn v_1 phi → Function.updateITE V v d v_1 = V v_1
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro v' a1	case s1.h1 Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D ⊢ ∀ (v_1 : VarName), isFreeIn v_1 phi → Function.updateITE V v d v_1 = V v_1	case s1.h1 Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi ⊢ Function.updateITE V v d v' = V v'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Function.updateITE]	case s1.h1 Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi ⊢ Function.updateITE V v d v' = V v'	case s1.h1 Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi ⊢ (if v' = v then d else V v') = V v'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	split_ifs	case s1.h1 Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi ⊢ (if v' = v then d else V v') = V v'	case pos Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi h✝ : v' = v ⊢ d = V v' case neg Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi h✝ : ¬v' = v ⊢ V v' = V v'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case _ c1 => subst c1 contradiction	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi c1 : v' = v ⊢ d = V v'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case _ c1 => rfl	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi c1 : ¬v' = v ⊢ V v' = V v'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	subst c1	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi c1 : v' = v ⊢ d = V v'	Δ : List Formula F phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi ih : ¬isFreeIn v' phi ⊢ d = V v'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	contradiction	Δ : List Formula F phi : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi ih : ¬isFreeIn v' phi ⊢ d = V v'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	rfl	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : Holds D I V E phi d : D v' : VarName a1 : isFreeIn v' phi c1 : ¬v' = v ⊢ V v' = V v'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F : Formula v : VarName phi : Formula a✝ : IsDeduct [] phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (forall_ v phi)	Δ : List Formula F : Formula v : VarName phi : Formula a✝¹ : IsDeduct [] phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (forall_ v phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F : Formula v : VarName phi : Formula a✝¹ : IsDeduct [] phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (forall_ v phi)	Δ : List Formula F : Formula v : VarName phi : Formula a✝¹ : IsDeduct [] phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ ∀ (d : D), Holds D I (Function.updateITE V v d) E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro d	Δ : List Formula F : Formula v : VarName phi : Formula a✝¹ : IsDeduct [] phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ ∀ (d : D), Holds D I (Function.updateITE V v d) E phi	Δ : List Formula F : Formula v : VarName phi : Formula a✝¹ : IsDeduct [] phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H d : D ⊢ Holds D I (Function.updateITE V v d) E phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	apply ih_2	Δ : List Formula F : Formula v : VarName phi : Formula a✝¹ : IsDeduct [] phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H d : D ⊢ Holds D I (Function.updateITE V v d) E phi	case a Δ : List Formula F : Formula v : VarName phi : Formula a✝¹ : IsDeduct [] phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H d : D ⊢ ∀ H ∈ [], Holds D I (Function.updateITE V v d) E H
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp	case a Δ : List Formula F : Formula v : VarName phi : Formula a✝¹ : IsDeduct [] phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H d : D ⊢ ∀ H ∈ [], Holds D I (Function.updateITE V v d) E H	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F : Formula v : VarName ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (eq_ v v)	Δ : List Formula F : Formula v : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (eq_ v v)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F : Formula v : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (eq_ v v)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1)))	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1)))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1)))	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ V x_0 = V y_0 ∧ V x_1 = V y_1 → (V x_0 = V x_1 ↔ V y_0 = V y_1)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro a2	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ V x_0 = V y_0 ∧ V x_1 = V y_1 → (V x_0 = V x_1 ↔ V y_0 = V y_1)	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : V x_0 = V y_0 ∧ V x_1 = V y_1 ⊢ V x_0 = V x_1 ↔ V y_0 = V y_1
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	cases a2	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2 : V x_0 = V y_0 ∧ V x_1 = V y_1 ⊢ V x_0 = V x_1 ↔ V y_0 = V y_1	case intro Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H left✝ : V x_0 = V y_0 right✝ : V x_1 = V y_1 ⊢ V x_0 = V x_1 ↔ V y_0 = V y_1
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case _ a2_left a2_right => simp only [a2_left] simp only [a2_right]	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2_left : V x_0 = V y_0 a2_right : V x_1 = V y_1 ⊢ V x_0 = V x_1 ↔ V y_0 = V y_1	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [a2_left]	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2_left : V x_0 = V y_0 a2_right : V x_1 = V y_1 ⊢ V x_0 = V x_1 ↔ V y_0 = V y_1	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2_left : V x_0 = V y_0 a2_right : V x_1 = V y_1 ⊢ V y_0 = V x_1 ↔ V y_0 = V y_1
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [a2_right]	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a2_left : V x_0 = V y_0 a2_right : V x_1 = V y_1 ⊢ V y_0 = V x_1 ↔ V y_0 = V y_1	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F : Formula X : PredName xs ys : List VarName ih_1 : xs.length = ys.length ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((List.foldr and_ true_ (List.zipWith eq_ xs ys)).imp_ ((pred_var_ X xs).iff_ (pred_var_ X ys)))	Δ : List Formula F : Formula X : PredName xs ys : List VarName ih_1 : xs.length = ys.length D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E ((List.foldr and_ true_ (List.zipWith eq_ xs ys)).imp_ ((pred_var_ X xs).iff_ (pred_var_ X ys)))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F : Formula X : PredName xs ys : List VarName ih_1 : xs.length = ys.length D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E ((List.foldr and_ true_ (List.zipWith eq_ xs ys)).imp_ ((pred_var_ X xs).iff_ (pred_var_ X ys)))	Δ : List Formula F : Formula X : PredName xs ys : List VarName ih_1 : xs.length = ys.length D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys)) → (I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V ys))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro a1	Δ : List Formula F : Formula X : PredName xs ys : List VarName ih_1 : xs.length = ys.length D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys)) → (I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V ys))	Δ : List Formula F : Formula X : PredName xs ys : List VarName ih_1 : xs.length = ys.length D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys)) ⊢ I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V ys)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	congr! 1	Δ : List Formula F : Formula X : PredName xs ys : List VarName ih_1 : xs.length = ys.length D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys)) ⊢ I.pred_var_ X (List.map V xs) ↔ I.pred_var_ X (List.map V ys)	case a.h.e'_4 Δ : List Formula F : Formula X : PredName xs ys : List VarName ih_1 : xs.length = ys.length D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys)) ⊢ List.map V xs = List.map V ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	induction xs generalizing ys	case a.h.e'_4 Δ : List Formula F : Formula X : PredName xs ys : List VarName ih_1 : xs.length = ys.length D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs ys)) ⊢ List.map V xs = List.map V ys	case a.h.e'_4.nil Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ys : List VarName ih_1 : [].length = ys.length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] ys)) ⊢ List.map V [] = List.map V ys case a.h.e'_4.cons Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H head✝ : VarName tail✝ : List VarName tail_ih✝ : ∀ (ys : List VarName), tail✝.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ tail✝ ys)) → List.map V tail✝ = List.map V ys ys : List VarName ih_1 : (head✝ :: tail✝).length = ys.length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (head✝ :: tail✝) ys)) ⊢ List.map V (head✝ :: tail✝) = List.map V ys
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case _ => cases ys case _ => simp only [Holds] case cons ys_hd ys_tl => simp at ih_1	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ys : List VarName ih_1 : [].length = ys.length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] ys)) ⊢ List.map V [] = List.map V ys	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	cases ys	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ys : List VarName ih_1 : [].length = ys.length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] ys)) ⊢ List.map V [] = List.map V ys	case nil Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ih_1 : [].length = [].length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] [])) ⊢ List.map V [] = List.map V [] case cons Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H head✝ : VarName tail✝ : List VarName ih_1 : [].length = (head✝ :: tail✝).length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] (head✝ :: tail✝))) ⊢ List.map V [] = List.map V (head✝ :: tail✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case _ => simp only [Holds]	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ih_1 : [].length = [].length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] [])) ⊢ List.map V [] = List.map V []	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case cons ys_hd ys_tl => simp at ih_1	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ys_hd : VarName ys_tl : List VarName ih_1 : [].length = (ys_hd :: ys_tl).length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] (ys_hd :: ys_tl))) ⊢ List.map V [] = List.map V (ys_hd :: ys_tl)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds]	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ih_1 : [].length = [].length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] [])) ⊢ List.map V [] = List.map V []	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp at ih_1	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ys_hd : VarName ys_tl : List VarName ih_1 : [].length = (ys_hd :: ys_tl).length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ [] (ys_hd :: ys_tl))) ⊢ List.map V [] = List.map V (ys_hd :: ys_tl)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	cases ys	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys : List VarName ih_1 : (xs_hd :: xs_tl).length = ys.length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) ys)) ⊢ List.map V (xs_hd :: xs_tl) = List.map V ys	case nil Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ih_1 : (xs_hd :: xs_tl).length = [].length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) [])) ⊢ List.map V (xs_hd :: xs_tl) = List.map V [] case cons Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys head✝ : VarName tail✝ : List VarName ih_1 : (xs_hd :: xs_tl).length = (head✝ :: tail✝).length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) (head✝ :: tail✝))) ⊢ List.map V (xs_hd :: xs_tl) = List.map V (head✝ :: tail✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case nil => simp at ih_1	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ih_1 : (xs_hd :: xs_tl).length = [].length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) [])) ⊢ List.map V (xs_hd :: xs_tl) = List.map V []	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp at ih_1	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ih_1 : (xs_hd :: xs_tl).length = [].length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) [])) ⊢ List.map V (xs_hd :: xs_tl) = List.map V []	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp at ih_1	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : (xs_hd :: xs_tl).length = (ys_hd :: ys_tl).length a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) (ys_hd :: ys_tl))) ⊢ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) (ys_hd :: ys_tl))) ih_1 : xs_tl.length = ys_tl.length ⊢ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp at a1	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName a1 : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ (xs_hd :: xs_tl) (ys_hd :: ys_tl))) ih_1 : xs_tl.length = ys_tl.length ⊢ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName a1 : Holds D I V E ((eq_ xs_hd ys_hd).and_ (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))) ih_1 : xs_tl.length = ys_tl.length ⊢ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp only [Holds] at a1	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName a1 : Holds D I V E ((eq_ xs_hd ys_hd).and_ (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl))) ih_1 : xs_tl.length = ys_tl.length ⊢ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : xs_tl.length = ys_tl.length a1 : V xs_hd = V ys_hd ∧ Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)) ⊢ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	cases a1	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : xs_tl.length = ys_tl.length a1 : V xs_hd = V ys_hd ∧ Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)) ⊢ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)	case intro Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : xs_tl.length = ys_tl.length left✝ : V xs_hd = V ys_hd right✝ : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)) ⊢ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	simp	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : xs_tl.length = ys_tl.length a1_left : V xs_hd = V ys_hd a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)) ⊢ List.map V (xs_hd :: xs_tl) = List.map V (ys_hd :: ys_tl)	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : xs_tl.length = ys_tl.length a1_left : V xs_hd = V ys_hd a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)) ⊢ V xs_hd = V ys_hd ∧ List.map V xs_tl = List.map V ys_tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	constructor	Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : xs_tl.length = ys_tl.length a1_left : V xs_hd = V ys_hd a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)) ⊢ V xs_hd = V ys_hd ∧ List.map V xs_tl = List.map V ys_tl	case left Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : xs_tl.length = ys_tl.length a1_left : V xs_hd = V ys_hd a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)) ⊢ V xs_hd = V ys_hd case right Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : xs_tl.length = ys_tl.length a1_left : V xs_hd = V ys_hd a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)) ⊢ List.map V xs_tl = List.map V ys_tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	exact a1_left	case left Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : xs_tl.length = ys_tl.length a1_left : V xs_hd = V ys_hd a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)) ⊢ V xs_hd = V ys_hd	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	apply xs_ih ys_tl ih_1 a1_right	case right Δ : List Formula F : Formula X : PredName D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H xs_hd : VarName xs_tl : List VarName xs_ih : ∀ (ys : List VarName), xs_tl.length = ys.length → Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys)) → List.map V xs_tl = List.map V ys ys_hd : VarName ys_tl : List VarName ih_1 : xs_tl.length = ys_tl.length a1_left : V xs_hd = V ys_hd a1_right : Holds D I V E (List.foldr and_ true_ (List.zipWith eq_ xs_tl ys_tl)) ⊢ List.map V xs_tl = List.map V ys_tl	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E _	Δ : List Formula F : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (false_.iff_ true_.not_)	Δ : List Formula F : Formula D : Type I : Interpretation D V : VarAssignment D E : Env a✝ : ∀ H ∈ [], Holds D I V E H ⊢ Holds D I V E (false_.iff_ true_.not_)

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.