file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
sequence | commit
stringclasses 4
values | url
stringclasses 4
values | start
sequence |
|---|---|---|---|---|---|---|
Mathlib/Data/Num/Lemmas.lean | Num.to_of_nat | [
{
"state_after": "no goals",
"state_before": "⊢ ↑↑0 = 0",
"tactic": "rw [Nat.cast_zero, cast_zero]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ↑↑(n + 1) = n + 1",
"tactic": "rw [Nat.cast_succ, add_one, succ_to_nat, to_of_nat n]"
}
] | [
492,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
490,
1
] |
Mathlib/Data/PNat/Interval.lean | PNat.card_fintype_Ico | [
{
"state_after": "no goals",
"state_before": "a b : ℕ+\n⊢ Fintype.card ↑(Set.Ico a b) = ↑b - ↑a",
"tactic": "rw [← card_Ico, Fintype.card_ofFinset]"
}
] | [
102,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
] |
Mathlib/RingTheory/IntegralClosure.lean | Algebra.IsIntegral.of_finite | [
{
"state_after": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.630412\nS : Type ?u.630415\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nh : Module.Finite R A\n⊢ RingHom.Finite (algebraMap R A)",
"state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.630412\nS : Type ?u.630415\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nh : Module.Finite R A\n⊢ Algebra.IsIntegral R A",
"tactic": "apply RingHom.Finite.to_isIntegral"
},
{
"state_after": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.630412\nS : Type ?u.630415\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nh : Module.Finite R A\n⊢ Module.Finite R A",
"state_before": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.630412\nS : Type ?u.630415\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nh : Module.Finite R A\n⊢ RingHom.Finite (algebraMap R A)",
"tactic": "rw [RingHom.Finite]"
},
{
"state_after": "case h.e'_5.h.e'_5\nR : Type u_1\nA : Type u_2\nB : Type ?u.630412\nS : Type ?u.630415\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nh : Module.Finite R A\n⊢ RingHom.toAlgebra (algebraMap R A) = inst✝¹",
"state_before": "case h\nR : Type u_1\nA : Type u_2\nB : Type ?u.630412\nS : Type ?u.630415\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nh : Module.Finite R A\n⊢ Module.Finite R A",
"tactic": "convert h"
},
{
"state_after": "case h.e'_5.h.e'_5.h\nR : Type u_1\nA : Type u_2\nB : Type ?u.630412\nS : Type ?u.630415\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nh : Module.Finite R A\nr✝ : R\nx✝ : A\n⊢ (let_fun I := RingHom.toAlgebra (algebraMap R A);\n r✝ • x✝) =\n r✝ • x✝",
"state_before": "case h.e'_5.h.e'_5\nR : Type u_1\nA : Type u_2\nB : Type ?u.630412\nS : Type ?u.630415\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nh : Module.Finite R A\n⊢ RingHom.toAlgebra (algebraMap R A) = inst✝¹",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.e'_5.h\nR : Type u_1\nA : Type u_2\nB : Type ?u.630412\nS : Type ?u.630415\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nh : Module.Finite R A\nr✝ : R\nx✝ : A\n⊢ (let_fun I := RingHom.toAlgebra (algebraMap R A);\n r✝ • x✝) =\n r✝ • x✝",
"tactic": "exact (Algebra.smul_def _ _).symm"
}
] | [
450,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
445,
1
] |
Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | differentiableWithinAt_iff_restrictScalars | [
{
"state_after": "case mp\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf' : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt 𝕜 f s x\nhs : UniqueDiffWithinAt 𝕜 s x\n⊢ DifferentiableWithinAt 𝕜' f s x → ∃ g', restrictScalars 𝕜 g' = fderivWithin 𝕜 f s x\n\ncase mpr\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf' : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt 𝕜 f s x\nhs : UniqueDiffWithinAt 𝕜 s x\n⊢ (∃ g', restrictScalars 𝕜 g' = fderivWithin 𝕜 f s x) → DifferentiableWithinAt 𝕜' f s x",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf' : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt 𝕜 f s x\nhs : UniqueDiffWithinAt 𝕜 s x\n⊢ DifferentiableWithinAt 𝕜' f s x ↔ ∃ g', restrictScalars 𝕜 g' = fderivWithin 𝕜 f s x",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf' : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt 𝕜 f s x\nhs : UniqueDiffWithinAt 𝕜 s x\ng' : E →L[𝕜'] F\nhg' : HasFDerivWithinAt f g' s x\n⊢ ∃ g', restrictScalars 𝕜 g' = fderivWithin 𝕜 f s x",
"state_before": "case mp\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf' : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt 𝕜 f s x\nhs : UniqueDiffWithinAt 𝕜 s x\n⊢ DifferentiableWithinAt 𝕜' f s x → ∃ g', restrictScalars 𝕜 g' = fderivWithin 𝕜 f s x",
"tactic": "rintro ⟨g', hg'⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf' : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt 𝕜 f s x\nhs : UniqueDiffWithinAt 𝕜 s x\ng' : E →L[𝕜'] F\nhg' : HasFDerivWithinAt f g' s x\n⊢ ∃ g', restrictScalars 𝕜 g' = fderivWithin 𝕜 f s x",
"tactic": "exact ⟨g', hs.eq (hg'.restrictScalars 𝕜) hf.hasFDerivWithinAt⟩"
},
{
"state_after": "case mpr.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf'✝ : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt 𝕜 f s x\nhs : UniqueDiffWithinAt 𝕜 s x\nf' : E →L[𝕜'] F\nhf' : restrictScalars 𝕜 f' = fderivWithin 𝕜 f s x\n⊢ DifferentiableWithinAt 𝕜' f s x",
"state_before": "case mpr\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf' : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt 𝕜 f s x\nhs : UniqueDiffWithinAt 𝕜 s x\n⊢ (∃ g', restrictScalars 𝕜 g' = fderivWithin 𝕜 f s x) → DifferentiableWithinAt 𝕜' f s x",
"tactic": "rintro ⟨f', hf'⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nf : E → F\nf'✝ : E →L[𝕜'] F\ns : Set E\nx : E\nhf : DifferentiableWithinAt 𝕜 f s x\nhs : UniqueDiffWithinAt 𝕜 s x\nf' : E →L[𝕜'] F\nhf' : restrictScalars 𝕜 f' = fderivWithin 𝕜 f s x\n⊢ DifferentiableWithinAt 𝕜' f s x",
"tactic": "exact ⟨f', hasFDerivWithinAt_of_restrictScalars 𝕜 hf.hasFDerivWithinAt hf'⟩"
}
] | [
116,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean | exists_associated_pow_of_mul_eq_pow | [
{
"state_after": "case inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Subsingleton α\n⊢ ∃ d, Associated (d ^ k) a\n\ncase inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\n⊢ ∃ d, Associated (d ^ k) a",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\n⊢ ∃ d, Associated (d ^ k) a",
"tactic": "cases subsingleton_or_nontrivial α"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : a = 0\n⊢ ∃ d, Associated (d ^ k) a\n\ncase neg\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\n⊢ ∃ d, Associated (d ^ k) a",
"state_before": "case inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\n⊢ ∃ d, Associated (d ^ k) a",
"tactic": "by_cases ha : a = 0"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : b = 0\n⊢ ∃ d, Associated (d ^ k) a\n\ncase neg\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\n⊢ ∃ d, Associated (d ^ k) a",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\n⊢ ∃ d, Associated (d ^ k) a",
"tactic": "by_cases hb : b = 0"
},
{
"state_after": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nh : a * b = c ^ 0\n⊢ ∃ d, Associated (d ^ 0) a\n\ncase neg.inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\n⊢ ∃ d, Associated (d ^ k) a",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\n⊢ ∃ d, Associated (d ^ k) a",
"tactic": "obtain rfl | hk := k.eq_zero_or_pos"
},
{
"state_after": "case neg.inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc : c ∣ a * b\n⊢ ∃ d, Associated (d ^ k) a",
"state_before": "case neg.inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\n⊢ ∃ d, Associated (d ^ k) a",
"tactic": "have hc : c ∣ a * b := by\n rw [h]\n exact dvd_pow_self _ hk.ne'"
},
{
"state_after": "case neg.inr.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₁ * d₂\n⊢ ∃ d, Associated (d ^ k) a",
"state_before": "case neg.inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc : c ∣ a * b\n⊢ ∃ d, Associated (d ^ k) a",
"tactic": "obtain ⟨d₁, d₂, hd₁, hd₂, hc⟩ := exists_dvd_and_dvd_of_dvd_mul hc"
},
{
"state_after": "case neg.inr.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₁ * d₂\n⊢ Associated (d₁ ^ k) a",
"state_before": "case neg.inr.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₁ * d₂\n⊢ ∃ d, Associated (d ^ k) a",
"tactic": "use d₁"
},
{
"state_after": "case neg.inr.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₁ * d₂\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\n⊢ Associated (d₁ ^ k) a",
"state_before": "case neg.inr.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₁ * d₂\n⊢ Associated (d₁ ^ k) a",
"tactic": "obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁"
},
{
"state_after": "case neg.inr.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : b * a = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\n⊢ Associated (d₁ ^ k) a",
"state_before": "case neg.inr.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₁ * d₂\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\n⊢ Associated (d₁ ^ k) a",
"tactic": "rw [mul_comm] at h hc"
},
{
"state_after": "case neg.inr.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh : b * a = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\n⊢ Associated (d₁ ^ k) a",
"state_before": "case neg.inr.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : b * a = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\n⊢ Associated (d₁ ^ k) a",
"tactic": "rw [(gcd_comm' a b).isUnit_iff] at hab"
},
{
"state_after": "case neg.inr.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh : b * a = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nhb' : b = d₂ ^ k * b'\n⊢ Associated (d₁ ^ k) a",
"state_before": "case neg.inr.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh : b * a = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\n⊢ Associated (d₁ ^ k) a",
"tactic": "obtain ⟨h0₂, ⟨b', hb'⟩⟩ := pow_dvd_of_mul_eq_pow hb hab h hc hd₂"
},
{
"state_after": "case neg.inr.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ Associated (d₁ ^ k) a",
"state_before": "case neg.inr.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh : b * a = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nhb' : b = d₂ ^ k * b'\n⊢ Associated (d₁ ^ k) a",
"tactic": "rw [ha', hb', hc, mul_pow] at h"
},
{
"state_after": "case neg.inr.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\nh' : a' * b' = 1\n⊢ Associated (d₁ ^ k) a",
"state_before": "case neg.inr.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ Associated (d₁ ^ k) a",
"tactic": "have h' : a' * b' = 1 := by\n apply (mul_right_inj' h0₁).mp\n rw [mul_one]\n apply (mul_right_inj' h0₂).mp\n rw [← h]\n rw [mul_assoc, mul_comm a', ← mul_assoc _ b', ← mul_assoc b', mul_comm b']"
},
{
"state_after": "case neg.inr.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\nh' : a' * b' = 1\n⊢ d₁ ^ k * ↑(Units.mkOfMulEqOne a' b' h') = a",
"state_before": "case neg.inr.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\nh' : a' * b' = 1\n⊢ Associated (d₁ ^ k) a",
"tactic": "use Units.mkOfMulEqOne _ _ h'"
},
{
"state_after": "no goals",
"state_before": "case neg.inr.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\nh' : a' * b' = 1\n⊢ d₁ ^ k * ↑(Units.mkOfMulEqOne a' b' h') = a",
"tactic": "rw [Units.val_mkOfMulEqOne, ha']"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Subsingleton α\n⊢ Associated (0 ^ k) a",
"state_before": "case inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Subsingleton α\n⊢ ∃ d, Associated (d ^ k) a",
"tactic": "use 0"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Subsingleton α\n⊢ Associated (0 ^ k) a",
"tactic": "rw [Subsingleton.elim a (0 ^ k)]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : a = 0\n⊢ Associated (0 ^ k) a",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : a = 0\n⊢ ∃ d, Associated (d ^ k) a",
"tactic": "use 0"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : a = 0\n⊢ Associated (0 ^ k) 0",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : a = 0\n⊢ Associated (0 ^ k) a",
"tactic": "rw [ha]"
},
{
"state_after": "case pos.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : a = 0\nh : a * b = c ^ 0\n⊢ Associated (0 ^ 0) 0\n\ncase pos.inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : a = 0\nhk : k > 0\n⊢ Associated (0 ^ k) 0",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : a = 0\n⊢ Associated (0 ^ k) 0",
"tactic": "obtain rfl | hk := k.eq_zero_or_pos"
},
{
"state_after": "case pos.inl.h\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : a = 0\nh : a * b = c ^ 0\n⊢ False",
"state_before": "case pos.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : a = 0\nh : a * b = c ^ 0\n⊢ Associated (0 ^ 0) 0",
"tactic": "exfalso"
},
{
"state_after": "case pos.inl.h\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : a = 0\n⊢ a * b = c ^ 0 → False",
"state_before": "case pos.inl.h\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : a = 0\nh : a * b = c ^ 0\n⊢ False",
"tactic": "revert h"
},
{
"state_after": "case pos.inl.h\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : a = 0\n⊢ 0 = 1 → False",
"state_before": "case pos.inl.h\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : a = 0\n⊢ a * b = c ^ 0 → False",
"tactic": "rw [ha, zero_mul, pow_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos.inl.h\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : a = 0\n⊢ 0 = 1 → False",
"tactic": "apply zero_ne_one"
},
{
"state_after": "no goals",
"state_before": "case pos.inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : a = 0\nhk : k > 0\n⊢ Associated (0 ^ k) 0",
"tactic": "rw [zero_pow hk]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : b = 0\n⊢ Associated (1 ^ k) a",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : b = 0\n⊢ ∃ d, Associated (d ^ k) a",
"tactic": "use 1"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : b = 0\n⊢ Associated 1 a",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : b = 0\n⊢ Associated (1 ^ k) a",
"tactic": "rw [one_pow]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : b = 0\n⊢ Associated (gcd a b) a",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : b = 0\n⊢ Associated 1 a",
"tactic": "apply (associated_one_iff_isUnit.mpr hab).symm.trans"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : b = 0\n⊢ Associated (gcd a 0) a",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : b = 0\n⊢ Associated (gcd a b) a",
"tactic": "rw [hb]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : b = 0\n⊢ Associated (gcd a 0) a",
"tactic": "exact gcd_zero_right' a"
},
{
"state_after": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nh : a * b = c ^ 0\n⊢ Associated (1 ^ 0) a",
"state_before": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nh : a * b = c ^ 0\n⊢ ∃ d, Associated (d ^ 0) a",
"tactic": "use 1"
},
{
"state_after": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nh : a * b = 1\n⊢ Associated 1 a",
"state_before": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nh : a * b = c ^ 0\n⊢ Associated (1 ^ 0) a",
"tactic": "rw [pow_zero] at h⊢"
},
{
"state_after": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nh : a * b = 1\n⊢ 1 * ↑(Units.mkOfMulEqOne a b h) = a",
"state_before": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nh : a * b = 1\n⊢ Associated 1 a",
"tactic": "use Units.mkOfMulEqOne _ _ h"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nh : a * b = 1\n⊢ 1 * ↑(Units.mkOfMulEqOne a b h) = a",
"tactic": "rw [Units.val_mkOfMulEqOne, one_mul]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\n⊢ c ∣ c ^ k",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\n⊢ c ∣ a * b",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nk : ℕ\nh : a * b = c ^ k\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\n⊢ c ∣ c ^ k",
"tactic": "exact dvd_pow_self _ hk.ne'"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ d₁ ^ k * (a' * b') = d₁ ^ k * 1",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ a' * b' = 1",
"tactic": "apply (mul_right_inj' h0₁).mp"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ d₁ ^ k * (a' * b') = d₁ ^ k",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ d₁ ^ k * (a' * b') = d₁ ^ k * 1",
"tactic": "rw [mul_one]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ d₂ ^ k * (d₁ ^ k * (a' * b')) = d₂ ^ k * d₁ ^ k",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ d₁ ^ k * (a' * b') = d₁ ^ k",
"tactic": "apply (mul_right_inj' h0₂).mp"
},
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ d₂ ^ k * (d₁ ^ k * (a' * b')) = d₂ ^ k * b' * (d₁ ^ k * a')",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ d₂ ^ k * (d₁ ^ k * (a' * b')) = d₂ ^ k * d₁ ^ k",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd b a)\nk : ℕ\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nhk : k > 0\nhc✝ : c ∣ a * b\nd₁ d₂ : α\nhd₁ : d₁ ∣ a\nhd₂ : d₂ ∣ b\nhc : c = d₂ * d₁\nh0₁ : d₁ ^ k ≠ 0\na' : α\nha' : a = d₁ ^ k * a'\nh0₂ : d₂ ^ k ≠ 0\nb' : α\nh : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k\nhb' : b = d₂ ^ k * b'\n⊢ d₂ ^ k * (d₁ ^ k * (a' * b')) = d₂ ^ k * b' * (d₁ ^ k * a')",
"tactic": "rw [mul_assoc, mul_comm a', ← mul_assoc _ b', ← mul_assoc b', mul_comm b']"
}
] | [
656,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
614,
1
] |
Mathlib/Logic/Function/Basic.lean | Function.update_comp_eq_of_forall_ne' | [] | [
621,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
619,
1
] |
Mathlib/Algebra/GeomSum.lean | mul_geom_sum | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Ring α\nx : α\nn : ℕ\n⊢ op ((x - 1) * ∑ i in range n, x ^ i) = op (x ^ n - 1)",
"tactic": "simpa using geom_sum_mul (op x) n"
}
] | [
226,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
225,
1
] |
Mathlib/RingTheory/Localization/NumDen.lean | IsFractionRing.mk'_num_den | [] | [
71,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean | CategoryTheory.Limits.HasColimit.mk | [] | [
649,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
648,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | InnerProductGeometry.norm_div_sin_angle_sub_of_inner_eq_zero | [
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\nh0 : x = 0 ∨ y ≠ 0\n⊢ ‖y‖ / Real.sin (angle x (x - y)) = ‖x - y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x = 0 ∨ y ≠ 0\n⊢ ‖y‖ / Real.sin (angle x (x - y)) = ‖x - y‖",
"tactic": "rw [← neg_eq_zero, ← inner_neg_right] at h"
},
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\nh0 : x = 0 ∨ -y ≠ 0\n⊢ ‖y‖ / Real.sin (angle x (x - y)) = ‖x - y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\nh0 : x = 0 ∨ y ≠ 0\n⊢ ‖y‖ / Real.sin (angle x (x - y)) = ‖x - y‖",
"tactic": "rw [← neg_ne_zero] at h0"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\nh0 : x = 0 ∨ -y ≠ 0\n⊢ ‖y‖ / Real.sin (angle x (x - y)) = ‖x - y‖",
"tactic": "rw [sub_eq_add_neg, ← norm_neg, norm_div_sin_angle_add_of_inner_eq_zero h h0]"
}
] | [
345,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
341,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.subtypeDomain_def | [
{
"state_after": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : Subtype p\n⊢ ↑(subtypeDomain p f) i = ↑(mk (Finset.subtype p (support f)) fun i => ↑f ↑↑i) i",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\n⊢ subtypeDomain p f = mk (Finset.subtype p (support f)) fun i => ↑f ↑↑i",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : Subtype p\n⊢ ↑(subtypeDomain p f) i = ↑(mk (Finset.subtype p (support f)) fun i => ↑f ↑↑i) i",
"tactic": "by_cases h2 : f i ≠ 0 <;> try simp at h2; dsimp; simp [h2]"
},
{
"state_after": "case neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : Subtype p\nh2 : ↑f ↑i = 0\n⊢ ↑(subtypeDomain p f) i = ↑(mk (Finset.subtype p (support f)) fun i => ↑f ↑↑i) i",
"state_before": "case neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : Subtype p\nh2 : ¬↑f ↑i ≠ 0\n⊢ ↑(subtypeDomain p f) i = ↑(mk (Finset.subtype p (support f)) fun i => ↑f ↑↑i) i",
"tactic": "simp at h2"
},
{
"state_after": "case neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : Subtype p\nh2 : ↑f ↑i = 0\n⊢ ↑f ↑i = if i ∈ Finset.subtype p (support f) then ↑f ↑i else 0",
"state_before": "case neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : Subtype p\nh2 : ↑f ↑i = 0\n⊢ ↑(subtypeDomain p f) i = ↑(mk (Finset.subtype p (support f)) fun i => ↑f ↑↑i) i",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\np : ι → Prop\ninst✝ : DecidablePred p\nf : Π₀ (i : ι), β i\ni : Subtype p\nh2 : ↑f ↑i = 0\n⊢ ↑f ↑i = if i ∈ Finset.subtype p (support f) then ↑f ↑i else 0",
"tactic": "simp [h2]"
}
] | [
1261,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1259,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | SubgroupClass.coe_inv | [] | [
229,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
228,
1
] |
Mathlib/Order/CompleteLattice.lean | iSup_sup_eq | [] | [
1234,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1232,
1
] |
Mathlib/Algebra/Module/Equiv.lean | LinearEquiv.toAddEquiv_toNatLinearEquiv | [] | [
818,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
816,
1
] |
Mathlib/Data/Finsupp/Order.lean | Finsupp.add_eq_zero_iff | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝ : CanonicallyOrderedAddMonoid α\nf g : ι →₀ α\n⊢ f + g = 0 ↔ f = 0 ∧ g = 0",
"tactic": "simp [FunLike.ext_iff, forall_and]"
}
] | [
154,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
153,
1
] |
Mathlib/Topology/Maps.lean | IsOpenMap.preimage_frontier_subset_frontier_preimage | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.196325\nδ : Type ?u.196328\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\nhf : IsOpenMap f\ns : Set β\n⊢ f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s)",
"tactic": "simpa only [frontier_eq_closure_inter_closure, preimage_inter] using\n inter_subset_inter hf.preimage_closure_subset_closure_preimage\n hf.preimage_closure_subset_closure_preimage"
}
] | [
434,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
430,
1
] |
Mathlib/Data/Set/Pairwise/Basic.lean | Set.pairwiseDisjoint_image_right_iff | [
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : α × β\nhx : x ∈ s ×ˢ t\ny : α × β\nhy : y ∈ s ×ˢ t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n⊢ x = y\n\ncase refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : InjOn (fun p => f p.fst p.snd) (s ×ˢ t)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nh : x ≠ y\n⊢ (Disjoint on fun a => f a '' t) x y",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\n⊢ (PairwiseDisjoint s fun a => f a '' t) ↔ InjOn (fun p => f p.fst p.snd) (s ×ˢ t)",
"tactic": "refine' ⟨fun hs x hx y hy (h : f _ _ = _) => _, fun hs x hx y hy h => _⟩"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : α × β\nhx : x ∈ s ×ˢ t\ny : α × β\nhy : y ∈ s ×ˢ t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n⊢ x.fst = y.fst",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : α × β\nhx : x ∈ s ×ˢ t\ny : α × β\nhy : y ∈ s ×ˢ t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n⊢ x = y",
"tactic": "suffices x.1 = y.1 by exact Prod.ext this (hf _ hx.1 <| h.trans <| by rw [this])"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : α × β\nhx : x ∈ s ×ˢ t\ny : α × β\nhy : y ∈ s ×ˢ t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n⊢ f x.fst x.snd ∈ f y.fst '' t",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : α × β\nhx : x ∈ s ×ˢ t\ny : α × β\nhy : y ∈ s ×ˢ t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n⊢ x.fst = y.fst",
"tactic": "refine' hs.elim hx.1 hy.1 (not_disjoint_iff.2 ⟨_, mem_image_of_mem _ hx.2, _⟩)"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : α × β\nhx : x ∈ s ×ˢ t\ny : α × β\nhy : y ∈ s ×ˢ t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n⊢ (fun p => f p.fst p.snd) y ∈ f y.fst '' t",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : α × β\nhx : x ∈ s ×ˢ t\ny : α × β\nhy : y ∈ s ×ˢ t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n⊢ f x.fst x.snd ∈ f y.fst '' t",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : α × β\nhx : x ∈ s ×ˢ t\ny : α × β\nhy : y ∈ s ×ˢ t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\n⊢ (fun p => f p.fst p.snd) y ∈ f y.fst '' t",
"tactic": "exact mem_image_of_mem _ hy.2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : α × β\nhx : x ∈ s ×ˢ t\ny : α × β\nhy : y ∈ s ×ˢ t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\nthis : x.fst = y.fst\n⊢ x = y",
"tactic": "exact Prod.ext this (hf _ hx.1 <| h.trans <| by rw [this])"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : PairwiseDisjoint s fun a => f a '' t\nx : α × β\nhx : x ∈ s ×ˢ t\ny : α × β\nhy : y ∈ s ×ˢ t\nh : f x.fst x.snd = (fun p => f p.fst p.snd) y\nthis : x.fst = y.fst\n⊢ (fun p => f p.fst p.snd) y = f x.fst y.snd",
"tactic": "rw [this]"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : InjOn (fun p => f p.fst p.snd) (s ×ˢ t)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nh : x ≠ y\n⊢ (fun a => f a '' t) x ⊓ (fun a => f a '' t) y ≤ ⊥",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : InjOn (fun p => f p.fst p.snd) (s ×ˢ t)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nh : x ≠ y\n⊢ (Disjoint on fun a => f a '' t) x y",
"tactic": "refine' disjoint_iff_inf_le.mpr _"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : InjOn (fun p => f p.fst p.snd) (s ×ˢ t)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nh : x ≠ y\na : β\nha : a ∈ t\nb : β\nhb : b ∈ t\nhab : f x a = f y b\n⊢ f y b ∈ ⊥",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : InjOn (fun p => f p.fst p.snd) (s ×ˢ t)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nh : x ≠ y\n⊢ (fun a => f a '' t) x ⊓ (fun a => f a '' t) y ≤ ⊥",
"tactic": "rintro _ ⟨⟨a, ha, hab⟩, b, hb, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type ?u.40777\nι' : Type ?u.40780\nr p q : α → α → Prop\nf : α → β → γ\ns : Set α\nt : Set β\nhf : ∀ (a : α), a ∈ s → Injective (f a)\nhs : InjOn (fun p => f p.fst p.snd) (s ×ˢ t)\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\nh : x ≠ y\na : β\nha : a ∈ t\nb : β\nhb : b ∈ t\nhab : f x a = f y b\n⊢ f y b ∈ ⊥",
"tactic": "exact h (congr_arg Prod.fst <| hs (mk_mem_prod hx ha) (mk_mem_prod hy hb) hab)"
}
] | [
376,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
366,
1
] |
Mathlib/ModelTheory/Basic.lean | FirstOrder.Language.card_relations_sum | [
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\ni : ℕ\n⊢ (#Relations (Language.sum L L') i) = lift (#Relations L i) + lift (#Relations L' i)",
"tactic": "simp [Language.sum]"
}
] | [
263,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
260,
1
] |
Mathlib/Topology/PartitionOfUnity.lean | BumpCovering.toPouFun_zero_of_zero | [
{
"state_after": "no goals",
"state_before": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : BumpCovering ι X s\ni : ι\nx : X\nh : ↑(toFun s f i) x = 0\n⊢ toPouFun f i x = 0",
"tactic": "rw [toPouFun, h, MulZeroClass.zero_mul]"
}
] | [
382,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
381,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean | mul_le_mul_of_nonneg_of_nonpos | [] | [
365,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
363,
1
] |
Mathlib/Data/List/Basic.lean | List.Sublist.antisymm | [] | [
1100,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1099,
1
] |
Mathlib/RingTheory/Subring/Basic.lean | Subring.mem_centralizer_iff | [] | [
866,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
865,
1
] |
Mathlib/Order/Hom/Lattice.lean | LatticeHom.coe_id | [] | [
1091,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1090,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean | Algebra.smul_leftMulMatrix | [
{
"state_after": "no goals",
"state_before": "R : Type u_3\nS : Type u_5\nT : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Ring T\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra S T\ninst✝⁵ : Algebra R T\ninst✝⁴ : IsScalarTower R S T\nm : Type u_1\nn : Type u_2\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : T\nik jk : m × n\n⊢ ↑(leftMulMatrix (Basis.smul b c)) x ik jk = ↑(leftMulMatrix b) (↑(leftMulMatrix c) x ik.snd jk.snd) ik.fst jk.fst",
"tactic": "simp only [leftMulMatrix_apply, LinearMap.toMatrix_apply, mul_comm, Basis.smul_apply,\n Basis.smul_repr, Finsupp.smul_apply, id.smul_eq_mul, LinearEquiv.map_smul, mul_smul_comm,\n coe_lmul_eq_mul, LinearMap.mul_apply']"
}
] | [
932,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
928,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | ContinuousOn.intervalIntegrable | [] | [
359,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
357,
1
] |
Mathlib/Algebra/Homology/Homology.lean | HomologicalComplex.cycles_eq_kernelSubobject | [] | [
56,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
54,
1
] |
Mathlib/Analysis/SumIntegralComparisons.lean | AntitoneOn.integral_le_sum | [
{
"state_after": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\n⊢ ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))",
"tactic": "intro k hk"
},
{
"state_after": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ uIcc (x₀ + ↑k) (x₀ + ↑(k + 1)) ⊆ Icc x₀ (x₀ + ↑a)",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))",
"tactic": "refine' (hf.mono _).intervalIntegrable"
},
{
"state_after": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ Icc (x₀ + ↑k) (x₀ + ↑(k + 1)) ⊆ Icc x₀ (x₀ + ↑a)\n\nx₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ x₀ + ↑k ≤ x₀ + ↑(k + 1)",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ uIcc (x₀ + ↑k) (x₀ + ↑(k + 1)) ⊆ Icc x₀ (x₀ + ↑a)",
"tactic": "rw [uIcc_of_le]"
},
{
"state_after": "case h₁\nx₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ x₀ ≤ x₀ + ↑k\n\ncase h₂\nx₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ x₀ + ↑(k + 1) ≤ x₀ + ↑a",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ Icc (x₀ + ↑k) (x₀ + ↑(k + 1)) ⊆ Icc x₀ (x₀ + ↑a)",
"tactic": "apply Icc_subset_Icc"
},
{
"state_after": "no goals",
"state_before": "case h₁\nx₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ x₀ ≤ x₀ + ↑k",
"tactic": "simp only [le_add_iff_nonneg_right, Nat.cast_nonneg]"
},
{
"state_after": "no goals",
"state_before": "case h₂\nx₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ x₀ + ↑(k + 1) ≤ x₀ + ↑a",
"tactic": "simp only [add_le_add_iff_left, Nat.cast_le, Nat.succ_le_of_lt hk]"
},
{
"state_after": "no goals",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nk : ℕ\nhk : k < a\n⊢ x₀ + ↑k ≤ x₀ + ↑(k + 1)",
"tactic": "simp only [add_le_add_iff_left, Nat.cast_le, Nat.le_succ]"
},
{
"state_after": "case h.e'_2.h.e'_6\nx₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\n⊢ x₀ = x₀ + ↑0",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\n⊢ (∫ (x : ℝ) in x₀..x₀ + ↑a, f x) = ∑ i in Finset.range a, ∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f x",
"tactic": "convert (intervalIntegral.sum_integral_adjacent_intervals hint).symm"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_6\nx₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\n⊢ x₀ = x₀ + ↑0",
"tactic": "simp only [Nat.cast_zero, add_zero]"
},
{
"state_after": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\n⊢ (∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f x) ≤ ∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f (x₀ + ↑i)",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\n⊢ (∑ i in Finset.range a, ∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f x) ≤\n ∑ i in Finset.range a, ∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f (x₀ + ↑i)",
"tactic": "apply Finset.sum_le_sum fun i hi => ?_"
},
{
"state_after": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\n⊢ (∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f x) ≤ ∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f (x₀ + ↑i)",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\n⊢ (∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f x) ≤ ∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f (x₀ + ↑i)",
"tactic": "have ia : i < a := Finset.mem_range.1 hi"
},
{
"state_after": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\nx : ℝ\nhx : x ∈ Icc (x₀ + ↑i) (x₀ + ↑(i + 1))\n⊢ f x ≤ f (x₀ + ↑i)",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\n⊢ (∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f x) ≤ ∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f (x₀ + ↑i)",
"tactic": "refine' intervalIntegral.integral_mono_on (by simp) (hint _ ia) (by simp) fun x hx => _"
},
{
"state_after": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\nx : ℝ\nhx : x ∈ Icc (x₀ + ↑i) (x₀ + ↑(i + 1))\n⊢ x₀ + ↑i ∈ Icc x₀ (x₀ + ↑a)\n\nx₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\nx : ℝ\nhx : x ∈ Icc (x₀ + ↑i) (x₀ + ↑(i + 1))\n⊢ x ∈ Icc x₀ (x₀ + ↑a)",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\nx : ℝ\nhx : x ∈ Icc (x₀ + ↑i) (x₀ + ↑(i + 1))\n⊢ f x ≤ f (x₀ + ↑i)",
"tactic": "apply hf _ _ hx.1"
},
{
"state_after": "no goals",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\n⊢ x₀ + ↑i ≤ x₀ + ↑(i + 1)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\n⊢ IntervalIntegrable (fun x => f (x₀ + ↑i)) volume (x₀ + ↑i) (x₀ + ↑(i + 1))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\nx : ℝ\nhx : x ∈ Icc (x₀ + ↑i) (x₀ + ↑(i + 1))\n⊢ x₀ + ↑i ∈ Icc x₀ (x₀ + ↑a)",
"tactic": "simp only [ia.le, mem_Icc, le_add_iff_nonneg_right, Nat.cast_nonneg, add_le_add_iff_left,\n Nat.cast_le, and_self_iff]"
},
{
"state_after": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\nx : ℝ\nhx : x ∈ Icc (x₀ + ↑i) (x₀ + ↑(i + 1))\n⊢ x₀ + ↑(i + 1) ≤ x₀ + ↑a",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\nx : ℝ\nhx : x ∈ Icc (x₀ + ↑i) (x₀ + ↑(i + 1))\n⊢ x ∈ Icc x₀ (x₀ + ↑a)",
"tactic": "refine' mem_Icc.2 ⟨le_trans (by simp) hx.1, le_trans hx.2 _⟩"
},
{
"state_after": "no goals",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\nx : ℝ\nhx : x ∈ Icc (x₀ + ↑i) (x₀ + ↑(i + 1))\n⊢ x₀ + ↑(i + 1) ≤ x₀ + ↑a",
"tactic": "simp only [add_le_add_iff_left, Nat.cast_le, Nat.succ_le_of_lt ia]"
},
{
"state_after": "no goals",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\ni : ℕ\nhi : i ∈ Finset.range a\nia : i < a\nx : ℝ\nhx : x ∈ Icc (x₀ + ↑i) (x₀ + ↑(i + 1))\n⊢ x₀ ≤ x₀ + ↑i",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "x₀ : ℝ\na b : ℕ\nf : ℝ → ℝ\nhf : AntitoneOn f (Icc x₀ (x₀ + ↑a))\nhint : ∀ (k : ℕ), k < a → IntervalIntegrable f volume (x₀ + ↑k) (x₀ + ↑(k + 1))\n⊢ (∑ i in Finset.range a, ∫ (x : ℝ) in x₀ + ↑i..x₀ + ↑(i + 1), f (x₀ + ↑i)) = ∑ i in Finset.range a, f (x₀ + ↑i)",
"tactic": "simp"
}
] | [
76,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.subset_coe_filter_of_subset_forall | [] | [
2746,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2745,
1
] |
Mathlib/Order/LiminfLimsup.lean | Filter.isBoundedUnder_ge_inv | [] | [
273,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
271,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.coe_filter | [] | [
1924,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1923,
1
] |
Mathlib/RingTheory/Filtration.lean | Ideal.Filtration.mem_submodule | [] | [
282,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
281,
1
] |
Mathlib/Data/MvPolynomial/Division.lean | MvPolynomial.monomial_dvd_monomial | [
{
"state_after": "case mp\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\n⊢ ↑(monomial i) r ∣ ↑(monomial j) s → (s = 0 ∨ i ≤ j) ∧ r ∣ s\n\ncase mpr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s → ↑(monomial i) r ∣ ↑(monomial j) s",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\n⊢ ↑(monomial i) r ∣ ↑(monomial j) s ↔ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ↑(monomial j) s = ↑(monomial i) r * x\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"state_before": "case mp\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\n⊢ ↑(monomial i) r ∣ ↑(monomial j) s → (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "rintro ⟨x, hx⟩"
},
{
"state_after": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"state_before": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ↑(monomial j) s = ↑(monomial i) r * x\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "rw [MvPolynomial.ext_iff] at hx"
},
{
"state_after": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhj : coeff j (↑(monomial j) s) = coeff j (↑(monomial i) r * x)\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"state_before": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "have hj := hx j"
},
{
"state_after": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhj : coeff j (↑(monomial j) s) = coeff j (↑(monomial i) r * x)\nhi : coeff i (↑(monomial j) s) = coeff i (↑(monomial i) r * x)\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"state_before": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhj : coeff j (↑(monomial j) s) = coeff j (↑(monomial i) r * x)\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "have hi := hx i"
},
{
"state_after": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi : (if j = i then s else 0) = coeff i (↑(monomial i) r * x)\nhj : s = coeff j (↑(monomial i) r * x)\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"state_before": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhj : coeff j (↑(monomial j) s) = coeff j (↑(monomial i) r * x)\nhi : coeff i (↑(monomial j) s) = coeff i (↑(monomial i) r * x)\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "simp_rw [coeff_monomial, if_pos] at hj hi"
},
{
"state_after": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0\nhj : s = if i ≤ j then r * coeff (j - i) x else 0\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"state_before": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi : (if j = i then s else 0) = coeff i (↑(monomial i) r * x)\nhj : s = coeff j (↑(monomial i) r * x)\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "simp_rw [coeff_monomial_mul'] at hi hj"
},
{
"state_after": "case mp.intro.inl.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi✝¹ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0\nhi✝ : j = i\nhi : i ≤ j\nhj : s = r * coeff (j - i) x\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s\n\ncase mp.intro.inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi✝¹ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0\nhi✝ : j = i\nhi : ¬i ≤ j\nhj : s = 0\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s\n\ncase mp.intro.inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi✝ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0\nhi : ¬j = i\nh✝ : i ≤ j\nhj : s = r * coeff (j - i) x\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s\n\ncase mp.intro.inr.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi✝ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0\nhi : ¬j = i\nh✝ : ¬i ≤ j\nhj : s = 0\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"state_before": "case mp.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0\nhj : s = if i ≤ j then r * coeff (j - i) x else 0\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "split_ifs at hi hj with hi hi"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.inl.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi✝¹ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0\nhi✝ : j = i\nhi : i ≤ j\nhj : s = r * coeff (j - i) x\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "exact ⟨Or.inr hi, _, hj⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi✝¹ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0\nhi✝ : j = i\nhi : ¬i ≤ j\nhj : s = 0\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "exact ⟨Or.inl hj, hj.symm ▸ dvd_zero _⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi✝ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0\nhi : ¬j = i\nh✝ : i ≤ j\nhj : s = r * coeff (j - i) x\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "simp_all only [or_true, dvd_mul_right]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.inr.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\nx : MvPolynomial σ R\nhx : ∀ (m : σ →₀ ℕ), coeff m (↑(monomial j) s) = coeff m (↑(monomial i) r * x)\nhi✝ : (if j = i then s else 0) = if i ≤ i then r * coeff (i - i) x else 0\nhi : ¬j = i\nh✝ : ¬i ≤ j\nhj : s = 0\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s",
"tactic": "simp_all only [ite_self, le_refl, ite_true, dvd_mul_right]"
},
{
"state_after": "case mpr.intro.inl.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\ni j : σ →₀ ℕ\nd : R\nh : r * d = 0\n⊢ ↑(monomial i) r ∣ ↑(monomial j) (r * d)\n\ncase mpr.intro.inr.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\ni j : σ →₀ ℕ\nhij : i ≤ j\nd : R\n⊢ ↑(monomial i) r ∣ ↑(monomial j) (r * d)",
"state_before": "case mpr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr s : R\ni j : σ →₀ ℕ\n⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s → ↑(monomial i) r ∣ ↑(monomial j) s",
"tactic": "rintro ⟨h | hij, d, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.inl.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\ni j : σ →₀ ℕ\nd : R\nh : r * d = 0\n⊢ ↑(monomial i) r ∣ ↑(monomial j) (r * d)",
"tactic": "simp_rw [h, monomial_zero, dvd_zero]"
},
{
"state_after": "case mpr.intro.inr.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\ni j : σ →₀ ℕ\nhij : i ≤ j\nd : R\n⊢ ↑(monomial j) (r * d) = ↑(monomial i) r * ↑(monomial (j - i)) d",
"state_before": "case mpr.intro.inr.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\ni j : σ →₀ ℕ\nhij : i ≤ j\nd : R\n⊢ ↑(monomial i) r ∣ ↑(monomial j) (r * d)",
"tactic": "refine' ⟨monomial (j - i) d, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.inr.intro\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nr : R\ni j : σ →₀ ℕ\nhij : i ≤ j\nd : R\n⊢ ↑(monomial j) (r * d) = ↑(monomial i) r * ↑(monomial (j - i)) d",
"tactic": "rw [monomial_mul, add_tsub_cancel_of_le hij]"
}
] | [
243,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
224,
1
] |
Mathlib/Analysis/Normed/Group/Completion.lean | UniformSpace.Completion.norm_coe | [] | [
39,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
38,
1
] |
Mathlib/Computability/PartrecCode.lean | Nat.Partrec.Code.evaln_prim | [
{
"state_after": "x✝ : Unit\np : ℕ\n⊢ List.map\n (fun n =>\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n (ofNat Code (unpair p).snd))\n (unpair p).fst)\n (List.range (unpair p).fst) =\n List.map (evaln (unpair p).fst (ofNat Code (unpair p).snd)) (List.range (unpair p).fst)",
"state_before": "x✝ : Unit\np : ℕ\n⊢ Nat.Partrec.Code.G\n (x✝,\n List.map\n (fun n =>\n let a := ofNat (ℕ × Code) n;\n List.map (evaln a.fst a.snd) (List.range a.fst))\n (List.range p)).snd =\n some\n (let a := ofNat (ℕ × Code) p;\n List.map (evaln a.fst a.snd) (List.range a.fst))",
"tactic": "simp only [G, prod_ofNat_val, ofNat_nat, List.length_map, List.length_range,\n Nat.pair_unpair, Option.some_inj]"
},
{
"state_after": "x✝ : Unit\np n : ℕ\n⊢ n ∈ List.range (unpair p).fst →\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n (ofNat Code (unpair p).snd))\n (unpair p).fst =\n evaln (unpair p).fst (ofNat Code (unpair p).snd) n",
"state_before": "x✝ : Unit\np : ℕ\n⊢ List.map\n (fun n =>\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n (ofNat Code (unpair p).snd))\n (unpair p).fst)\n (List.range (unpair p).fst) =\n List.map (evaln (unpair p).fst (ofNat Code (unpair p).snd)) (List.range (unpair p).fst)",
"tactic": "refine List.map_congr fun n => ?_"
},
{
"state_after": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\n⊢ n ∈ List.range (unpair p).fst →\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n (ofNat Code (unpair p).snd))\n (unpair p).fst =\n evaln (unpair p).fst (ofNat Code (unpair p).snd) n",
"state_before": "x✝ : Unit\np n : ℕ\n⊢ n ∈ List.range (unpair p).fst →\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n (ofNat Code (unpair p).snd))\n (unpair p).fst =\n evaln (unpair p).fst (ofNat Code (unpair p).snd) n",
"tactic": "have : List.range p = List.range (Nat.pair p.unpair.1 (encode (ofNat Code p.unpair.2))) := by\n simp"
},
{
"state_after": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\n⊢ n ∈ List.range (unpair p).fst →\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n (ofNat Code (unpair p).snd))\n (unpair p).fst =\n evaln (unpair p).fst (ofNat Code (unpair p).snd) n",
"state_before": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\n⊢ n ∈ List.range (unpair p).fst →\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n ((unpair p).fst, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range p))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n (ofNat Code (unpair p).snd))\n (unpair p).fst =\n evaln (unpair p).fst (ofNat Code (unpair p).snd) n",
"tactic": "rw [this]"
},
{
"state_after": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk : ℕ\n⊢ n ∈ List.range k →\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n (ofNat Code (unpair p).snd))\n k =\n evaln k (ofNat Code (unpair p).snd) n",
"state_before": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\n⊢ n ∈ List.range (unpair p).fst →\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n ((unpair p).fst, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n (ofNat Code (unpair p).snd))\n (unpair p).fst =\n evaln (unpair p).fst (ofNat Code (unpair p).snd) n",
"tactic": "generalize p.unpair.1 = k"
},
{
"state_after": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk : ℕ\nc : Code\n⊢ n ∈ List.range k →\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c)\n k =\n evaln k c n",
"state_before": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk : ℕ\n⊢ n ∈ List.range k →\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (k, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n (ofNat Code (unpair p).snd))\n k =\n evaln k (ofNat Code (unpair p).snd) n",
"tactic": "generalize ofNat Code p.unpair.2 = c"
},
{
"state_after": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk : ℕ\nc : Code\nnk : n ∈ List.range k\n⊢ Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c)\n k =\n evaln k c n",
"state_before": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk : ℕ\nc : Code\n⊢ n ∈ List.range k →\n Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c)\n k =\n evaln k c n",
"tactic": "intro nk"
},
{
"state_after": "case zero\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nnk : n ∈ List.range Nat.zero\n⊢ Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c)\n Nat.zero =\n evaln Nat.zero c n\n\ncase succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nnk : n ∈ List.range (Nat.succ k')\n⊢ Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c)\n (Nat.succ k') =\n evaln (Nat.succ k') c n",
"state_before": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk : ℕ\nc : Code\nnk : n ∈ List.range k\n⊢ Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c)\n k =\n evaln k c n",
"tactic": "cases' k with k'"
},
{
"state_after": "case succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nnk : n ∈ List.range (Nat.succ k')\nk : ℕ := k' + 1\n⊢ Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c)\n (Nat.succ k') =\n evaln (Nat.succ k') c n",
"state_before": "case succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nnk : n ∈ List.range (Nat.succ k')\n⊢ Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c)\n (Nat.succ k') =\n evaln (Nat.succ k') c n",
"tactic": "let k := k' + 1"
},
{
"state_after": "case succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nnk : n ∈ List.range (Nat.succ k')\nk : ℕ := k' + 1\n⊢ rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) (unpair n).fst)\n (fun n_1 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst n_1)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c =\n evaln (k' + 1) c n",
"state_before": "case succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nnk : n ∈ List.range (Nat.succ k')\nk : ℕ := k' + 1\n⊢ Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (Nat.succ k', cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (Nat.succ k') (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c)\n (Nat.succ k') =\n evaln (Nat.succ k') c n",
"tactic": "simp only [show k'.succ = k from rfl]"
},
{
"state_after": "case succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\n⊢ rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) (unpair n).fst)\n (fun n_1 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst n_1)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c =\n evaln (k' + 1) c n",
"state_before": "case succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nnk : n ∈ List.range (Nat.succ k')\nk : ℕ := k' + 1\n⊢ rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) (unpair n).fst)\n (fun n_1 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst n_1)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c =\n evaln (k' + 1) c n",
"tactic": "simp [Nat.lt_succ_iff] at nk"
},
{
"state_after": "case succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode c) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k', c') n =\n evaln k' c' n\n⊢ rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) (unpair n).fst)\n (fun n_1 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst n_1)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c =\n evaln (k' + 1) c n",
"state_before": "case succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\n⊢ rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) (unpair n).fst)\n (fun n_1 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst n_1)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c =\n evaln (k' + 1) c n",
"tactic": "have hg :\n ∀ {k' c' n},\n Nat.pair k' (encode c') < Nat.pair k (encode c) →\n lup ((List.range (Nat.pair k (encode c))).map fun n =>\n (List.range n.unpair.1).map (evaln n.unpair.1 (ofNat Code n.unpair.2))) (k', c') n =\n evaln k' c' n := by\n intro k₁ c₁ n₁ hl\n simp [lup, List.get?_range hl, evaln_map, Bind.bind]"
},
{
"state_after": "case succ.pair\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (pair cf cg)))))\n (k', c') n =\n evaln k' c' n\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n (k' + 1, cf) n)\n fun x =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n (k' + 1, cg) n)\n fun y => some (Nat.pair x y)) =\n Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)\n\ncase succ.comp\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (comp cf cg)))))\n (k', c') n =\n evaln k' c' n\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cg) n)\n fun x =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cf) x) =\n Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x\n\ncase succ.prec\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\n⊢ Nat.rec\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cf) (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd\n\ncase succ.rfind'\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k' + 1, cf) n)\n fun x =>\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x) =\n Option.bind (evaln (k' + 1) cf n) fun x =>\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"state_before": "case succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode c) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k', c') n =\n evaln k' c' n\n⊢ rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) n\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) (unpair n).fst)\n (fun n_1 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst n_1)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k' + 1, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode c))))\n (k', c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c =\n evaln (k' + 1) c n",
"tactic": "cases' c with cf cg cf cg cf cg cf <;>\n simp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]"
},
{
"state_after": "no goals",
"state_before": "x✝ : Unit\np n : ℕ\n⊢ List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case zero\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nnk : n ∈ List.range Nat.zero\n⊢ Nat.rec Option.none\n (fun n_1 n_ih =>\n rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cf) n\n let y ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cg) n\n some (Nat.pair x y))\n (fun cf cg x x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cg) n\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cf) x)\n (fun cf cg x x =>\n Nat.rec\n (Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cf) (unpair n).fst)\n (fun n_2 n_ih => do\n let i ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n =>\n List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst n_2)\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n (unpair n).snd)\n (fun cf x => do\n let x ←\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (Nat.zero, cf) n\n Nat.rec (some (unpair n).snd)\n (fun n_2 n_ih =>\n Nat.Partrec.Code.lup\n (List.map\n (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair Nat.zero (encode c))))\n (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x)\n c)\n Nat.zero =\n evaln Nat.zero c n",
"tactic": "simp [evaln]"
},
{
"state_after": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\nk₁ : ℕ\nc₁ : Code\nn₁ : ℕ\nhl : Nat.pair k₁ (encode c₁) < Nat.pair k (encode c)\n⊢ Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k₁, c₁) n₁ =\n evaln k₁ c₁ n₁",
"state_before": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\n⊢ ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode c) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k', c') n =\n evaln k' c' n",
"tactic": "intro k₁ c₁ n₁ hl"
},
{
"state_after": "no goals",
"state_before": "x✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\nk₁ : ℕ\nc₁ : Code\nn₁ : ℕ\nhl : Nat.pair k₁ (encode c₁) < Nat.pair k (encode c)\n⊢ Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode c))))\n (k₁, c₁) n₁ =\n evaln k₁ c₁ n₁",
"tactic": "simp [lup, List.get?_range hl, evaln_map, Bind.bind]"
},
{
"state_after": "case succ.pair.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (pair cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n (k' + 1, cf) n)\n fun x =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n (k' + 1, cg) n)\n fun y => some (Nat.pair x y)) =\n Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)",
"state_before": "case succ.pair\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (pair cf cg)))))\n (k', c') n =\n evaln k' c' n\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n (k' + 1, cf) n)\n fun x =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n (k' + 1, cg) n)\n fun y => some (Nat.pair x y)) =\n Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)",
"tactic": "cases' encode_lt_pair cf cg with lf lg"
},
{
"state_after": "case succ.pair.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (pair cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n⊢ (Option.bind (evaln k cf n) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)",
"state_before": "case succ.pair.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (pair cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n (k' + 1, cf) n)\n fun x =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n (k' + 1, cg) n)\n fun y => some (Nat.pair x y)) =\n Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)",
"tactic": "rw [hg (Nat.pair_lt_pair_right _ lf), hg (Nat.pair_lt_pair_right _ lg)]"
},
{
"state_after": "case succ.pair.intro.none\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (pair cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n⊢ (Option.bind Option.none fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n Option.bind (Option.map Nat.pair Option.none) fun y => Option.map y (evaln (k' + 1) cg n)\n\ncase succ.pair.intro.some\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (pair cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\nval✝ : ℕ\n⊢ (Option.bind (some val✝) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n Option.bind (Option.map Nat.pair (some val✝)) fun y => Option.map y (evaln (k' + 1) cg n)",
"state_before": "case succ.pair.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (pair cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n⊢ (Option.bind (evaln k cf n) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)",
"tactic": "cases evaln k cf n"
},
{
"state_after": "no goals",
"state_before": "case succ.pair.intro.some\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (pair cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\nval✝ : ℕ\n⊢ (Option.bind (some val✝) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n Option.bind (Option.map Nat.pair (some val✝)) fun y => Option.map y (evaln (k' + 1) cg n)",
"tactic": "cases evaln k cg n <;> rfl"
},
{
"state_after": "no goals",
"state_before": "case succ.pair.intro.none\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (pair cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n⊢ (Option.bind Option.none fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n Option.bind (Option.map Nat.pair Option.none) fun y => Option.map y (evaln (k' + 1) cg n)",
"tactic": "rfl"
},
{
"state_after": "case succ.comp.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (comp cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cg) n)\n fun x =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cf) x) =\n Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x",
"state_before": "case succ.comp\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (comp cf cg)))))\n (k', c') n =\n evaln k' c' n\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cg) n)\n fun x =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cf) x) =\n Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x",
"tactic": "cases' encode_lt_comp cf cg with lf lg"
},
{
"state_after": "case succ.comp.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (comp cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n⊢ (Option.bind (evaln k cg n) fun x =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cf) x) =\n Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x",
"state_before": "case succ.comp.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (comp cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cg) n)\n fun x =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cf) x) =\n Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x",
"tactic": "rw [hg (Nat.pair_lt_pair_right _ lg)]"
},
{
"state_after": "case succ.comp.intro.none\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (comp cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n⊢ (Option.bind Option.none fun x =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cf) x) =\n Option.bind Option.none fun x => evaln (k' + 1) cf x\n\ncase succ.comp.intro.some\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (comp cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\nval✝ : ℕ\n⊢ (Option.bind (some val✝) fun x =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cf) x) =\n Option.bind (some val✝) fun x => evaln (k' + 1) cf x",
"state_before": "case succ.comp.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (comp cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n⊢ (Option.bind (evaln k cg n) fun x =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cf) x) =\n Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x",
"tactic": "cases evaln k cg n"
},
{
"state_after": "no goals",
"state_before": "case succ.comp.intro.some\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (comp cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\nval✝ : ℕ\n⊢ (Option.bind (some val✝) fun x =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cf) x) =\n Option.bind (some val✝) fun x => evaln (k' + 1) cf x",
"tactic": "simp [hg (Nat.pair_lt_pair_right _ lf)]"
},
{
"state_after": "no goals",
"state_before": "case succ.comp.intro.none\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (comp cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n⊢ (Option.bind Option.none fun x =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n (k' + 1, cf) x) =\n Option.bind Option.none fun x => evaln (k' + 1) cf x",
"tactic": "rfl"
},
{
"state_after": "case succ.prec.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n⊢ Nat.rec\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cf) (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd",
"state_before": "case succ.prec\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\n⊢ Nat.rec\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cf) (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd",
"tactic": "cases' encode_lt_prec cf cg with lf lg"
},
{
"state_after": "case succ.prec.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n⊢ Nat.rec (evaln k cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd",
"state_before": "case succ.prec.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n⊢ Nat.rec\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cf) (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd",
"tactic": "rw [hg (Nat.pair_lt_pair_right _ lf)]"
},
{
"state_after": "case succ.prec.intro.zero\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n⊢ Nat.rec (evaln k cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n Nat.zero =\n Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n Nat.zero\n\ncase succ.prec.intro.succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn✝ : ℕ\n⊢ Nat.rec (evaln k cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (Nat.succ n✝) =\n Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (Nat.succ n✝)",
"state_before": "case succ.prec.intro\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n⊢ Nat.rec (evaln k cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd =\n Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (unpair n).snd",
"tactic": "cases n.unpair.2"
},
{
"state_after": "case succ.prec.intro.succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn✝ : ℕ\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n✝))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n✝ i))) =\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n✝)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n✝ i))",
"state_before": "case succ.prec.intro.succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn✝ : ℕ\n⊢ Nat.rec (evaln k cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (Nat.succ n✝) =\n Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n (Nat.succ n✝)",
"tactic": "simp"
},
{
"state_after": "case succ.prec.intro.succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn✝ : ℕ\n⊢ (Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n✝)) fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n✝ i))) =\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n✝)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n✝ i))",
"state_before": "case succ.prec.intro.succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn✝ : ℕ\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n✝))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n✝ i))) =\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n✝)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n✝ i))",
"tactic": "rw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)]"
},
{
"state_after": "case succ.prec.intro.succ.none\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn✝ : ℕ\n⊢ (Option.bind Option.none fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n✝ i))) =\n Option.bind Option.none fun i => evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n✝ i))\n\ncase succ.prec.intro.succ.some\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn✝ val✝ : ℕ\n⊢ (Option.bind (some val✝) fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n✝ i))) =\n Option.bind (some val✝) fun i => evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n✝ i))",
"state_before": "case succ.prec.intro.succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn✝ : ℕ\n⊢ (Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n✝)) fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n✝ i))) =\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n✝)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n✝ i))",
"tactic": "cases evaln k' _ _"
},
{
"state_after": "no goals",
"state_before": "case succ.prec.intro.succ.some\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn✝ val✝ : ℕ\n⊢ (Option.bind (some val✝) fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n✝ i))) =\n Option.bind (some val✝) fun i => evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n✝ i))",
"tactic": "simp [hg (Nat.pair_lt_pair_right _ lg)]"
},
{
"state_after": "no goals",
"state_before": "case succ.prec.intro.zero\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n⊢ Nat.rec (evaln k cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n Nat.zero =\n Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n (fun n_1 n_ih =>\n Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n Nat.zero",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case succ.prec.intro.succ.none\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (prec cf cg)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn✝ : ℕ\n⊢ (Option.bind Option.none fun i =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n✝ i))) =\n Option.bind Option.none fun i => evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n✝ i))",
"tactic": "rfl"
},
{
"state_after": "case succ.rfind'\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k' + 1, cf) n)\n fun x =>\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x) =\n Option.bind (evaln (k' + 1) cf n) fun x =>\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"state_before": "case succ.rfind'\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k' + 1, cf) n)\n fun x =>\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x) =\n Option.bind (evaln (k' + 1) cf n) fun x =>\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"tactic": "have lf := encode_lt_rfind' cf"
},
{
"state_after": "case succ.rfind'\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n⊢ (Option.bind (evaln k cf n) fun x =>\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x) =\n Option.bind (evaln (k' + 1) cf n) fun x =>\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"state_before": "case succ.rfind'\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n⊢ (Option.bind\n (Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k' + 1, cf) n)\n fun x =>\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x) =\n Option.bind (evaln (k' + 1) cf n) fun x =>\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"tactic": "rw [hg (Nat.pair_lt_pair_right _ lf)]"
},
{
"state_after": "case succ.rfind'.none\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n⊢ (Option.bind Option.none fun x =>\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x) =\n Option.bind Option.none fun x =>\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n\ncase succ.rfind'.some\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nx : ℕ\n⊢ (Option.bind (some x) fun x =>\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x) =\n Option.bind (some x) fun x =>\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"state_before": "case succ.rfind'\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n⊢ (Option.bind (evaln k cf n) fun x =>\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x) =\n Option.bind (evaln (k' + 1) cf n) fun x =>\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"tactic": "cases' evaln k cf n with x"
},
{
"state_after": "case succ.rfind'.some\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nx : ℕ\n⊢ Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x =\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"state_before": "case succ.rfind'.some\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nx : ℕ\n⊢ (Option.bind (some x) fun x =>\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x) =\n Option.bind (some x) fun x =>\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"tactic": "simp"
},
{
"state_after": "case succ.rfind'.some.succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nn✝ : ℕ\n⊢ Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) =\n evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"state_before": "case succ.rfind'.some\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nx : ℕ\n⊢ Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x =\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"tactic": "cases x <;> simp [Nat.succ_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case succ.rfind'.some.succ\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nn✝ : ℕ\n⊢ Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) =\n evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"tactic": "rw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)]"
},
{
"state_after": "no goals",
"state_before": "case succ.rfind'.none\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) →\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair k (encode (rfind' cf)))))\n (k', c') n =\n evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n⊢ (Option.bind Option.none fun x =>\n Nat.rec (some (unpair n).snd)\n (fun n_1 n_ih =>\n Nat.Partrec.Code.lup\n (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n x) =\n Option.bind Option.none fun x =>\n if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "this :\n Primrec₂ fun x n =>\n let a := ofNat (ℕ × Code) n;\n List.map (evaln a.fst a.snd) (List.range a.fst)\nx✝ : (ℕ × Code) × ℕ\nk : ℕ\nc : Code\nn : ℕ\n⊢ (Option.bind\n (List.get?\n (let a := ofNat (ℕ × Code) (encode ((k, c), n).fst);\n List.map (evaln a.fst a.snd) (List.range a.fst))\n ((k, c), n).snd)\n fun b => (((k, c), n), b).snd) =\n evaln ((k, c), n).fst.fst ((k, c), n).fst.snd ((k, c), n).snd",
"tactic": "simp [evaln_map]"
}
] | [
1146,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1087,
1
] |
Mathlib/Logic/Equiv/List.lean | Denumerable.raise_sorted | [] | [
324,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
322,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean | LieSubalgebra.ext_iff | [] | [
177,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
176,
1
] |
Mathlib/Data/Nat/Dist.lean | Nat.dist_mul_left | [
{
"state_after": "no goals",
"state_before": "k n m : ℕ\n⊢ dist (k * n) (k * m) = k * dist n m",
"tactic": "rw [mul_comm k n, mul_comm k m, dist_mul_right, mul_comm]"
}
] | [
108,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
107,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean | CategoryTheory.GrothendieckTopology.Cover.Relation.map_fst | [] | [
550,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
548,
1
] |
Mathlib/Order/Minimal.lean | maximals_swap | [] | [
77,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
76,
1
] |
Std/Data/Int/DivMod.lean | Int.dvd_refl | [] | [
595,
74
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
595,
11
] |
Mathlib/GroupTheory/OrderOfElement.lean | orderOf_pow_dvd | [
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn✝ m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\nn : ℕ\n⊢ orderOf (x ^ n) ∣ orderOf x",
"tactic": "rw [orderOf_dvd_iff_pow_eq_one, pow_right_comm, pow_orderOf_eq_one, one_pow]"
}
] | [
254,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
253,
1
] |
Mathlib/NumberTheory/VonMangoldt.lean | Nat.ArithmeticFunction.vonMangoldt_sum | [
{
"state_after": "case refine'_1\nn : ℕ\n⊢ ∑ i in divisors 0, ↑Λ i = Real.log ↑0\n\ncase refine'_2\nn : ℕ\n⊢ ∀ (p n : ℕ), Prime p → ∑ i in divisors (p ^ n), ↑Λ i = Real.log ↑(p ^ n)\n\ncase refine'_3\nn : ℕ\n⊢ ∀ (a b : ℕ),\n 1 < a →\n 1 < b →\n coprime a b →\n ∑ i in divisors a, ↑Λ i = Real.log ↑a →\n ∑ i in divisors b, ↑Λ i = Real.log ↑b → ∑ i in divisors (a * b), ↑Λ i = Real.log ↑(a * b)",
"state_before": "n : ℕ\n⊢ ∑ i in divisors n, ↑Λ i = Real.log ↑n",
"tactic": "refine' recOnPrimeCoprime _ _ _ n"
},
{
"state_after": "case refine'_3\nn a b : ℕ\nha' : 1 < a\nhb' : 1 < b\nhab : coprime a b\nha : ∑ i in divisors a, ↑Λ i = Real.log ↑a\nhb : ∑ i in divisors b, ↑Λ i = Real.log ↑b\n⊢ ∑ i in divisors (a * b), ↑Λ i = Real.log ↑(a * b)",
"state_before": "case refine'_3\nn : ℕ\n⊢ ∀ (a b : ℕ),\n 1 < a →\n 1 < b →\n coprime a b →\n ∑ i in divisors a, ↑Λ i = Real.log ↑a →\n ∑ i in divisors b, ↑Λ i = Real.log ↑b → ∑ i in divisors (a * b), ↑Λ i = Real.log ↑(a * b)",
"tactic": "intro a b ha' hb' hab ha hb"
},
{
"state_after": "case refine'_3\nn a b : ℕ\nha' : 1 < a\nhb' : 1 < b\nhab : coprime a b\nha : ∑ a in filter (fun a => IsPrimePow a) (divisors a), Real.log ↑(minFac a) = Real.log ↑a\nhb : ∑ a in filter (fun a => IsPrimePow a) (divisors b), Real.log ↑(minFac a) = Real.log ↑b\n⊢ ∑ a in filter (fun a => IsPrimePow a) (divisors (a * b)), Real.log ↑(minFac a) = Real.log ↑(a * b)",
"state_before": "case refine'_3\nn a b : ℕ\nha' : 1 < a\nhb' : 1 < b\nhab : coprime a b\nha : ∑ i in divisors a, ↑Λ i = Real.log ↑a\nhb : ∑ i in divisors b, ↑Λ i = Real.log ↑b\n⊢ ∑ i in divisors (a * b), ↑Λ i = Real.log ↑(a * b)",
"tactic": "simp only [vonMangoldt_apply, ← sum_filter] at ha hb⊢"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nn a b : ℕ\nha' : 1 < a\nhb' : 1 < b\nhab : coprime a b\nha : ∑ a in filter (fun a => IsPrimePow a) (divisors a), Real.log ↑(minFac a) = Real.log ↑a\nhb : ∑ a in filter (fun a => IsPrimePow a) (divisors b), Real.log ↑(minFac a) = Real.log ↑b\n⊢ ∑ a in filter (fun a => IsPrimePow a) (divisors (a * b)), Real.log ↑(minFac a) = Real.log ↑(a * b)",
"tactic": "rw [mul_divisors_filter_prime_pow hab, filter_union,\n sum_union (disjoint_divisors_filter_isPrimePow hab), ha, hb, Nat.cast_mul,\n Real.log_mul (cast_ne_zero.2 (pos_of_gt ha').ne') (cast_ne_zero.2 (pos_of_gt hb').ne')]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nn : ℕ\n⊢ ∑ i in divisors 0, ↑Λ i = Real.log ↑0",
"tactic": "simp"
},
{
"state_after": "case refine'_2\nn p k : ℕ\nhp : Prime p\n⊢ ∑ i in divisors (p ^ k), ↑Λ i = Real.log ↑(p ^ k)",
"state_before": "case refine'_2\nn : ℕ\n⊢ ∀ (p n : ℕ), Prime p → ∑ i in divisors (p ^ n), ↑Λ i = Real.log ↑(p ^ n)",
"tactic": "intro p k hp"
},
{
"state_after": "case refine'_2\nn p k : ℕ\nhp : Prime p\n⊢ ∑ k in range k, ↑Λ (p ^ (k + 1)) + 0 = ↑k * Real.log ↑p",
"state_before": "case refine'_2\nn p k : ℕ\nhp : Prime p\n⊢ ∑ i in divisors (p ^ k), ↑Λ i = Real.log ↑(p ^ k)",
"tactic": "rw [sum_divisors_prime_pow hp, cast_pow, Real.log_pow, Finset.sum_range_succ', pow_zero,\n vonMangoldt_apply_one]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nn p k : ℕ\nhp : Prime p\n⊢ ∑ k in range k, ↑Λ (p ^ (k + 1)) + 0 = ↑k * Real.log ↑p",
"tactic": "simp [vonMangoldt_apply_pow (Nat.succ_ne_zero _), vonMangoldt_apply_prime hp]"
}
] | [
123,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
] |
Mathlib/Algebra/Order/SMul.lean | strictMono_smul_left | [] | [
142,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
141,
1
] |
Std/Data/Int/Lemmas.lean | Int.add_neg_of_nonpos_of_neg | [] | [
841,
52
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
840,
11
] |
Mathlib/Order/Hom/CompleteLattice.lean | CompleteLatticeHom.symm_dual_id | [] | [
895,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
893,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | CategoryTheory.Limits.Multicofork.sigma_condition | [
{
"state_after": "case h\nC : Type u\ninst✝² : Category C\nI : MultispanIndex C\nK : Multicofork I\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nb✝ : I.L\n⊢ Sigma.ι I.left b✝ ≫ MultispanIndex.fstSigmaMap I ≫ Sigma.desc (π K) =\n Sigma.ι I.left b✝ ≫ MultispanIndex.sndSigmaMap I ≫ Sigma.desc (π K)",
"state_before": "C : Type u\ninst✝² : Category C\nI : MultispanIndex C\nK : Multicofork I\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\n⊢ MultispanIndex.fstSigmaMap I ≫ Sigma.desc (π K) = MultispanIndex.sndSigmaMap I ≫ Sigma.desc (π K)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u\ninst✝² : Category C\nI : MultispanIndex C\nK : Multicofork I\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nb✝ : I.L\n⊢ Sigma.ι I.left b✝ ≫ MultispanIndex.fstSigmaMap I ≫ Sigma.desc (π K) =\n Sigma.ι I.left b✝ ≫ MultispanIndex.sndSigmaMap I ≫ Sigma.desc (π K)",
"tactic": "simp"
}
] | [
610,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
608,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean | SimpleGraph.Colorable.mono_left | [] | [
322,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
320,
1
] |
Mathlib/Data/Set/Image.lean | Set.range_inl | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.78850\nι : Sort ?u.78853\nι' : Sort ?u.78856\nf : ι → α\ns t : Set α\n⊢ range Sum.inl = {x | Sum.isLeft x = true}",
"tactic": "ext (_|_) <;> simp"
}
] | [
884,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
884,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean | Subsemigroup.eq_top_iff' | [] | [
946,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
945,
1
] |
Mathlib/Algebra/Hom/Group.lean | MonoidHom.ext_iff | [] | [
765,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
764,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean | AlgEquiv.symm_trans_apply | [] | [
407,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
405,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean | LinearPMap.map_sub | [] | [
109,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
] |
Mathlib/CategoryTheory/Adhesive.lean | CategoryTheory.Adhesive.van_kampen' | [] | [
236,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean | Localization.mk_left_injective | [
{
"state_after": "α : Type u_1\ninst✝ : CancelCommMonoid α\ns : Submonoid α\na₁ b₁ : α\na₂ b₂ b : { x // x ∈ s }\nc d : α\nh : (fun a => mk a b) c = (fun a => mk a b) d\nthis : Nonempty { x // x ∈ s }\n⊢ c = d",
"state_before": "α : Type u_1\ninst✝ : CancelCommMonoid α\ns : Submonoid α\na₁ b₁ : α\na₂ b₂ b : { x // x ∈ s }\nc d : α\nh : (fun a => mk a b) c = (fun a => mk a b) d\n⊢ c = d",
"tactic": "have : Nonempty s := One.nonempty"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CancelCommMonoid α\ns : Submonoid α\na₁ b₁ : α\na₂ b₂ b : { x // x ∈ s }\nc d : α\nh : (fun a => mk a b) c = (fun a => mk a b) d\nthis : Nonempty { x // x ∈ s }\n⊢ c = d",
"tactic": "simpa [-mk_eq_monoidOf_mk', mk_eq_mk_iff, r_iff_exists] using h"
}
] | [
1889,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1886,
1
] |
Mathlib/Topology/Algebra/Module/Multilinear.lean | ContinuousMultilinearMap.neg_apply | [] | [
438,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
437,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | Real.log_neg_eq_log | [
{
"state_after": "no goals",
"state_before": "x✝ y x : ℝ\n⊢ log (-x) = log x",
"tactic": "rw [← log_abs x, ← log_abs (-x), abs_neg]"
}
] | [
114,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
114,
1
] |
Mathlib/Deprecated/Subgroup.lean | Group.subset_closure | [] | [
544,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
544,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | Left.pow_lt_one_iff | [] | [
349,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
348,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | CategoryTheory.Limits.pushout.congrHom_inv | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ f₂ ≫ 𝟙 Y = 𝟙 X ≫ f₁",
"tactic": "simp [h₁]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ g₂ ≫ 𝟙 Z = 𝟙 X ≫ g₁",
"tactic": "simp [h₂]"
},
{
"state_after": "case h₀\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inl ≫ (congrHom h₁ h₂).inv =\n inl ≫ map f₂ g₂ f₁ g₁ (𝟙 Y) (𝟙 Z) (𝟙 X) (_ : f₂ ≫ 𝟙 Y = 𝟙 X ≫ f₁) (_ : g₂ ≫ 𝟙 Z = 𝟙 X ≫ g₁)\n\ncase h₁\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inr ≫ (congrHom h₁ h₂).inv =\n inr ≫ map f₂ g₂ f₁ g₁ (𝟙 Y) (𝟙 Z) (𝟙 X) (_ : f₂ ≫ 𝟙 Y = 𝟙 X ≫ f₁) (_ : g₂ ≫ 𝟙 Z = 𝟙 X ≫ g₁)",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ (congrHom h₁ h₂).inv = map f₂ g₂ f₁ g₁ (𝟙 Y) (𝟙 Z) (𝟙 X) (_ : f₂ ≫ 𝟙 Y = 𝟙 X ≫ f₁) (_ : g₂ ≫ 𝟙 Z = 𝟙 X ≫ g₁)",
"tactic": "ext"
},
{
"state_after": "case h₀\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inl ≫ (congrHom h₁ h₂).inv = 𝟙 Y ≫ inl",
"state_before": "case h₀\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inl ≫ (congrHom h₁ h₂).inv =\n inl ≫ map f₂ g₂ f₁ g₁ (𝟙 Y) (𝟙 Z) (𝟙 X) (_ : f₂ ≫ 𝟙 Y = 𝟙 X ≫ f₁) (_ : g₂ ≫ 𝟙 Z = 𝟙 X ≫ g₁)",
"tactic": "erw [pushout.inl_desc]"
},
{
"state_after": "case h₀\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inl = inl ≫ (congrHom h₁ h₂).hom",
"state_before": "case h₀\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inl ≫ (congrHom h₁ h₂).inv = 𝟙 Y ≫ inl",
"tactic": "rw [Iso.comp_inv_eq, Category.id_comp]"
},
{
"state_after": "case h₀\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inl = 𝟙 Y ≫ inl",
"state_before": "case h₀\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inl = inl ≫ (congrHom h₁ h₂).hom",
"tactic": "erw [pushout.inl_desc]"
},
{
"state_after": "no goals",
"state_before": "case h₀\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inl = 𝟙 Y ≫ inl",
"tactic": "rw [Category.id_comp]"
},
{
"state_after": "case h₁\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inr ≫ (congrHom h₁ h₂).inv = 𝟙 Z ≫ inr",
"state_before": "case h₁\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inr ≫ (congrHom h₁ h₂).inv =\n inr ≫ map f₂ g₂ f₁ g₁ (𝟙 Y) (𝟙 Z) (𝟙 X) (_ : f₂ ≫ 𝟙 Y = 𝟙 X ≫ f₁) (_ : g₂ ≫ 𝟙 Z = 𝟙 X ≫ g₁)",
"tactic": "erw [pushout.inr_desc]"
},
{
"state_after": "case h₁\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inr = inr ≫ (congrHom h₁ h₂).hom",
"state_before": "case h₁\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inr ≫ (congrHom h₁ h₂).inv = 𝟙 Z ≫ inr",
"tactic": "rw [Iso.comp_inv_eq, Category.id_comp]"
},
{
"state_after": "case h₁\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inr = 𝟙 Z ≫ inr",
"state_before": "case h₁\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inr = inr ≫ (congrHom h₁ h₂).hom",
"tactic": "erw [pushout.inr_desc]"
},
{
"state_after": "no goals",
"state_before": "case h₁\nC : Type u\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nW X✝ Y✝ Z✝ X Y Z : C\nf₁ f₂ : X ⟶ Y\ng₁ g₂ : X ⟶ Z\nh₁ : f₁ = f₂\nh₂ : g₁ = g₂\ninst✝¹ : HasPushout f₁ g₁\ninst✝ : HasPushout f₂ g₂\n⊢ inr = 𝟙 Z ≫ inr",
"tactic": "rw [Category.id_comp]"
}
] | [
1420,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1408,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean | EuclideanGeometry.subset_sphere | [] | [
107,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
1
] |
Mathlib/Topology/MetricSpace/Completion.lean | Isometry.completion_map | [] | [
215,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | HasDerivAt.cpow_const | [] | [
190,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/MeasureTheory/PiSystem.lean | isPiSystem_Ioc | [] | [
194,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
] |
Mathlib/Computability/Primrec.lean | Primrec.comp | [
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nσ : Type u_2\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf : β → σ\ng : α → β\nhf : Primrec f\nhg : Primrec g\nn : ℕ\n⊢ (Nat.casesOn (encode (decode n)) 0 fun n =>\n encode (Option.map f (decode (Nat.pred (encode (Option.map g (decode n))))))) =\n encode (Option.map (fun a => f (g a)) (decode n))",
"tactic": "cases @decode α _ n <;> simp [encodek]"
}
] | [
264,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
262,
1
] |
Mathlib/Data/Finset/Image.lean | Finset.image_symmDiff | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.94769\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt✝ : Finset β\na : α\nb c : β\ninst✝ : DecidableEq α\nf : α → β\ns t : Finset α\nhf : Injective f\n⊢ f '' ↑s ∆ ↑t = (f '' ↑s) ∆ (f '' ↑t)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.94769\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt✝ : Finset β\na : α\nb c : β\ninst✝ : DecidableEq α\nf : α → β\ns t : Finset α\nhf : Injective f\n⊢ ↑(image f (s ∆ t)) = ↑(image f s ∆ image f t)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.94769\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝ : Finset α\nt✝ : Finset β\na : α\nb c : β\ninst✝ : DecidableEq α\nf : α → β\ns t : Finset α\nhf : Injective f\n⊢ f '' ↑s ∆ ↑t = (f '' ↑s) ∆ (f '' ↑t)",
"tactic": "exact Set.image_symm_diff hf _ _"
}
] | [
542,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
538,
1
] |
src/lean/Init/SimpLemmas.lean | ite_self | [
{
"state_after": "no goals",
"state_before": "α : Sort u\nc : Prop\nd : Decidable c\na : α\n⊢ ite c a a = a",
"tactic": "cases d <;> rfl"
}
] | [
81,
113
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
81,
9
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.sub_coe_pi_eq_add_coe_pi | [
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ θ - ↑π = θ + ↑π",
"tactic": "rw [sub_eq_add_neg, neg_coe_pi]"
}
] | [
161,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean | toIcoMod_add_zsmul | [
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ b + m • p - (toIcoDiv hp a b • p + m • p) = b - toIcoDiv hp a b • p",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ toIcoMod hp a (b + m • p) = toIcoMod hp a b",
"tactic": "rw [toIcoMod, toIcoDiv_add_zsmul, toIcoMod, add_smul]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ b + m • p - (toIcoDiv hp a b • p + m • p) = b - toIcoDiv hp a b • p",
"tactic": "abel"
}
] | [
407,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
405,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean | Associates.dvd_out_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizationMonoid α\na : α\nb : Associates α\n⊢ ∀ (a_1 : α),\n a ∣ Associates.out (Quotient.mk (Associated.setoid α) a_1) ↔ Associates.mk a ≤ Quotient.mk (Associated.setoid α) a_1",
"tactic": "simp [Associates.out_mk, Associates.quotient_mk_eq_mk, mk_le_mk_iff_dvd_iff]"
}
] | [
238,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
236,
1
] |
Mathlib/Analysis/Convex/Gauge.lean | gauge_ball | [
{
"state_after": "case symmetric\n𝕜 : Type ?u.286698\nE : Type u_1\nF : Type ?u.286704\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nr : ℝ\nx✝ : E\nhr : 0 < r\nx : E\n⊢ ∀ (x : E), x ∈ Metric.ball 0 1 → -x ∈ Metric.ball 0 1",
"state_before": "𝕜 : Type ?u.286698\nE : Type u_1\nF : Type ?u.286704\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nr : ℝ\nx✝ : E\nhr : 0 < r\nx : E\n⊢ gauge (Metric.ball 0 r) x = ‖x‖ / r",
"tactic": "rw [← smul_unitBall_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,\n abs_of_nonneg hr.le, div_eq_inv_mul]"
},
{
"state_after": "case symmetric\n𝕜 : Type ?u.286698\nE : Type u_1\nF : Type ?u.286704\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nr : ℝ\nx✝ : E\nhr : 0 < r\nx : E\n⊢ ∀ (x : E), ‖x‖ < 1 → ‖x‖ < 1",
"state_before": "case symmetric\n𝕜 : Type ?u.286698\nE : Type u_1\nF : Type ?u.286704\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nr : ℝ\nx✝ : E\nhr : 0 < r\nx : E\n⊢ ∀ (x : E), x ∈ Metric.ball 0 1 → -x ∈ Metric.ball 0 1",
"tactic": "simp_rw [mem_ball_zero_iff, norm_neg]"
},
{
"state_after": "no goals",
"state_before": "case symmetric\n𝕜 : Type ?u.286698\nE : Type u_1\nF : Type ?u.286704\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nr : ℝ\nx✝ : E\nhr : 0 < r\nx : E\n⊢ ∀ (x : E), ‖x‖ < 1 → ‖x‖ < 1",
"tactic": "exact fun _ => id"
}
] | [
464,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
460,
1
] |
Mathlib/Algebra/Hom/Iterate.lean | RingHom.iterate_map_add | [] | [
140,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
] |
Mathlib/Algebra/Parity.lean | Even.mul_right | [] | [
287,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
286,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | QuadraticForm.associated_comp | [
{
"state_after": "case H\nS : Type u_3\nR : Type u_1\nR₁ : Type ?u.408596\nM : Type u_2\ninst✝⁹ : Ring R\ninst✝⁸ : CommRing R₁\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Module R₁ M\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra S R\ninst✝² : Invertible 2\nB₁ : BilinForm R M\nQ : QuadraticForm R M\nN : Type v\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : N →ₗ[R] M\nx✝ y✝ : N\n⊢ bilin (↑(associatedHom S) (comp Q f)) x✝ y✝ = bilin (BilinForm.comp (↑(associatedHom S) Q) f f) x✝ y✝",
"state_before": "S : Type u_3\nR : Type u_1\nR₁ : Type ?u.408596\nM : Type u_2\ninst✝⁹ : Ring R\ninst✝⁸ : CommRing R₁\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Module R₁ M\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra S R\ninst✝² : Invertible 2\nB₁ : BilinForm R M\nQ : QuadraticForm R M\nN : Type v\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : N →ₗ[R] M\n⊢ ↑(associatedHom S) (comp Q f) = BilinForm.comp (↑(associatedHom S) Q) f f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H\nS : Type u_3\nR : Type u_1\nR₁ : Type ?u.408596\nM : Type u_2\ninst✝⁹ : Ring R\ninst✝⁸ : CommRing R₁\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Module R₁ M\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra S R\ninst✝² : Invertible 2\nB₁ : BilinForm R M\nQ : QuadraticForm R M\nN : Type v\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : N →ₗ[R] M\nx✝ y✝ : N\n⊢ bilin (↑(associatedHom S) (comp Q f)) x✝ y✝ = bilin (BilinForm.comp (↑(associatedHom S) Q) f f) x✝ y✝",
"tactic": "simp only [QuadraticForm.comp_apply, BilinForm.comp_apply, associated_apply, LinearMap.map_add]"
}
] | [
793,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
790,
1
] |
Mathlib/Order/WellFoundedSet.lean | Set.isWf_empty | [] | [
192,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | QuadraticForm.polar_neg_left | [
{
"state_after": "no goals",
"state_before": "S : Type ?u.154687\nR : Type u_1\nR₁ : Type ?u.154693\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : CommRing R₁\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx y : M\n⊢ polar (↑Q) (-x) y = -polar (↑Q) x y",
"tactic": "rw [← neg_one_smul R x, polar_smul_left, neg_one_mul]"
}
] | [
278,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/Combinatorics/SimpleGraph/Metric.lean | SimpleGraph.dist_le | [] | [
69,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
68,
1
] |
Mathlib/Topology/UniformSpace/CompleteSeparated.lean | DenseInducing.continuous_extend_of_cauchy | [] | [
48,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
46,
1
] |
Mathlib/Algebra/Lie/Solvable.lean | LieIdeal.derivedSeries_eq_bot_iff | [
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nk : ℕ\n⊢ derivedSeries R { x // x ∈ ↑I } k = ⊥ ↔ derivedSeriesOfIdeal R L k I = ⊥",
"tactic": "rw [← derivedSeries_eq_derivedSeriesOfIdeal_map, map_eq_bot_iff, ker_incl, eq_bot_iff]"
}
] | [
171,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
] |
Mathlib/Data/Set/Basic.lean | Set.Subsingleton.anti | [] | [
2337,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2336,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.lintegral_add | [] | [
1046,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1045,
1
] |
Mathlib/Data/Int/Parity.lean | Int.ediv_two_mul_two_add_one_of_odd | [
{
"state_after": "case intro\nm c : ℤ\n⊢ (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1",
"state_before": "m n : ℤ\n⊢ Odd n → n / 2 * 2 + 1 = n",
"tactic": "rintro ⟨c, rfl⟩"
},
{
"state_after": "case h.e'_2.h.e'_6\nm c : ℤ\n⊢ 1 = (2 * c + 1) % 2",
"state_before": "case intro\nm c : ℤ\n⊢ (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1",
"tactic": "convert Int.ediv_add_emod' (2 * c + 1) 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_6\nm c : ℤ\n⊢ 1 = (2 * c + 1) % 2",
"tactic": "simp [Int.add_emod]"
}
] | [
270,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
267,
1
] |
Mathlib/LinearAlgebra/Matrix/Trace.lean | Matrix.trace_smul | [] | [
65,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.monotone_filter_right | [] | [
2737,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2735,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.natDegree_le_natDegree | [] | [
237,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/CategoryTheory/Abelian/Homology.lean | homology.hom_from_ext | [
{
"state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\n⊢ a = b",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh : π' f g w ≫ a = π' f g w ≫ b\n⊢ a = b",
"tactic": "dsimp [π'] at h"
},
{
"state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\n⊢ (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) a = (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) b\n\ncase inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\n⊢ Function.Injective fun e => (homologyIsoCokernelLift f g w).inv ≫ e",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\n⊢ a = b",
"tactic": "apply_fun fun e => (homologyIsoCokernelLift f g w).inv ≫ e"
},
{
"state_after": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\n⊢ Function.Injective fun e => (homologyIsoCokernelLift f g w).inv ≫ e\n\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\n⊢ (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) a = (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) b",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\n⊢ (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) a = (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) b\n\ncase inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\n⊢ Function.Injective fun e => (homologyIsoCokernelLift f g w).inv ≫ e",
"tactic": "swap"
},
{
"state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv ≫ a =\n cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv ≫ b\n⊢ (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) a = (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) b",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\n⊢ (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) a = (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) b",
"tactic": "simp only [Category.assoc] at h"
},
{
"state_after": "no goals",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv ≫ a =\n cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv ≫ b\n⊢ (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) a = (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) b",
"tactic": "exact coequalizer.hom_ext h"
},
{
"state_after": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\ni j : homology f g w ⟶ W\nhh : (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) i = (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) j\n⊢ i = j",
"state_before": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\n⊢ Function.Injective fun e => (homologyIsoCokernelLift f g w).inv ≫ e",
"tactic": "intro i j hh"
},
{
"state_after": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\ni j : homology f g w ⟶ W\nhh :\n (homologyIsoCokernelLift f g w).hom ≫ (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) i =\n (homologyIsoCokernelLift f g w).hom ≫ (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) j\n⊢ i = j",
"state_before": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\ni j : homology f g w ⟶ W\nhh : (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) i = (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) j\n⊢ i = j",
"tactic": "apply_fun fun e => (homologyIsoCokernelLift f g w).hom ≫ e at hh"
},
{
"state_after": "no goals",
"state_before": "case inj\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nW : A\na b : homology f g w ⟶ W\nh :\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ a =\n (cokernel.π (kernel.lift g f w) ≫ (homologyIsoCokernelLift f g w).inv) ≫ b\ni j : homology f g w ⟶ W\nhh :\n (homologyIsoCokernelLift f g w).hom ≫ (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) i =\n (homologyIsoCokernelLift f g w).hom ≫ (fun e => (homologyIsoCokernelLift f g w).inv ≫ e) j\n⊢ i = j",
"tactic": "simpa using hh"
}
] | [
175,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
166,
1
] |
Mathlib/GroupTheory/GroupAction/Prod.lean | Prod.pow_snd | [] | [
99,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
98,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | MeasureTheory.Subsingleton.stronglyMeasurable | [
{
"state_after": "case refine_2\nα✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nf_sf : α →ₛ β :=\n { toFun := f, measurableSet_fiber' := (_ : ∀ (x : β), MeasurableSet (f ⁻¹' {x})),\n finite_range' := (_ : Set.Finite (range f)) }\n⊢ StronglyMeasurable f\n\ncase refine_1\nα✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nx : β\n⊢ MeasurableSet (f ⁻¹' {x})",
"state_before": "α✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\n⊢ StronglyMeasurable f",
"tactic": "let f_sf : α →ₛ β := ⟨f, fun x => ?_, Set.Subsingleton.finite Set.subsingleton_of_subsingleton⟩"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nf_sf : α →ₛ β :=\n { toFun := f, measurableSet_fiber' := (_ : ∀ (x : β), MeasurableSet (f ⁻¹' {x})),\n finite_range' := (_ : Set.Finite (range f)) }\n⊢ StronglyMeasurable f",
"tactic": "exact ⟨fun _ => f_sf, fun x => tendsto_const_nhds⟩"
},
{
"state_after": "case refine_1\nα✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nx : β\nh_univ : f ⁻¹' {x} = univ\n⊢ MeasurableSet (f ⁻¹' {x})",
"state_before": "case refine_1\nα✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nx : β\n⊢ MeasurableSet (f ⁻¹' {x})",
"tactic": "have h_univ : f ⁻¹' {x} = Set.univ := by\n ext1 y\n simp"
},
{
"state_after": "case refine_1\nα✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nx : β\nh_univ : f ⁻¹' {x} = univ\n⊢ MeasurableSet univ",
"state_before": "case refine_1\nα✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nx : β\nh_univ : f ⁻¹' {x} = univ\n⊢ MeasurableSet (f ⁻¹' {x})",
"tactic": "rw [h_univ]"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nx : β\nh_univ : f ⁻¹' {x} = univ\n⊢ MeasurableSet univ",
"tactic": "exact MeasurableSet.univ"
},
{
"state_after": "case h\nα✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nx : β\ny : α\n⊢ y ∈ f ⁻¹' {x} ↔ y ∈ univ",
"state_before": "α✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nx : β\n⊢ f ⁻¹' {x} = univ",
"tactic": "ext1 y"
},
{
"state_after": "no goals",
"state_before": "case h\nα✝ : Type ?u.13198\nβ✝ : Type ?u.13201\nγ : Type ?u.13204\nι : Type ?u.13207\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton β\nf : α → β\nx : β\ny : α\n⊢ y ∈ f ⁻¹' {x} ↔ y ∈ univ",
"tactic": "simp"
}
] | [
147,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
] |
Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.BoundedFormula.realize_all_liftAt_one_self | [
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\n⊢ Realize (∀'liftAt 1 n φ) v xs ↔ Realize φ v xs",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\n⊢ Realize (∀'liftAt 1 n φ) v xs ↔ Realize φ v xs",
"tactic": "inhabit M"
},
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\n⊢ (∀ (a : M), Realize φ v (snoc xs a ∘ ↑castSucc)) ↔ Realize φ v xs",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\n⊢ Realize (∀'liftAt 1 n φ) v xs ↔ Realize φ v xs",
"tactic": "simp only [realize_all, realize_liftAt_one_self]"
},
{
"state_after": "case refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\nh : ∀ (a : M), Realize φ v (snoc xs a ∘ ↑castSucc)\n⊢ Realize φ v xs\n\ncase refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\nh : Realize φ v xs\na : M\n⊢ Realize φ v (snoc xs a ∘ ↑castSucc)",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\n⊢ (∀ (a : M), Realize φ v (snoc xs a ∘ ↑castSucc)) ↔ Realize φ v xs",
"tactic": "refine' ⟨fun h => _, fun h a => _⟩"
},
{
"state_after": "case refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\nh : ∀ (a : M), Realize φ v (snoc xs a ∘ ↑castSucc)\ni : Fin n\n⊢ (snoc xs default ∘ ↑castSucc) i = xs i",
"state_before": "case refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\nh : ∀ (a : M), Realize φ v (snoc xs a ∘ ↑castSucc)\n⊢ Realize φ v xs",
"tactic": "refine' (congr rfl (funext fun i => _)).mp (h default)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\nh : ∀ (a : M), Realize φ v (snoc xs a ∘ ↑castSucc)\ni : Fin n\n⊢ (snoc xs default ∘ ↑castSucc) i = xs i",
"tactic": "simp"
},
{
"state_after": "case refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\nh : Realize φ v xs\na : M\ni : Fin n\n⊢ xs i = (snoc xs a ∘ ↑castSucc) i",
"state_before": "case refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\nh : Realize φ v xs\na : M\n⊢ Realize φ v (snoc xs a ∘ ↑castSucc)",
"tactic": "refine' (congr rfl (funext fun i => _)).mp h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.165049\nP : Type ?u.165052\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\nα : Type u'\nβ : Type v'\nn✝ l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv✝ : α → M\nxs✝ : Fin l → M\ninst✝ : Nonempty M\nn : ℕ\nφ : BoundedFormula L α n\nv : α → M\nxs : Fin n → M\ninhabited_h : Inhabited M\nh : Realize φ v xs\na : M\ni : Fin n\n⊢ xs i = (snoc xs a ∘ ↑castSucc) i",
"tactic": "simp"
}
] | [
509,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
501,
1
] |
Mathlib/CategoryTheory/Preadditive/Biproducts.lean | CategoryTheory.Limits.inr_of_isLimit | [
{
"state_after": "case x\nC : Type u\ninst✝² : Category C\ninst✝¹ : Preadditive C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nt : BinaryBicone X Y\nht : IsLimit (BinaryBicone.toCone t)\n⊢ ∀ (j : Discrete WalkingPair), t.inr ≫ (BinaryBicone.toCone t).π.app j = (BinaryFan.mk 0 (𝟙 Y)).π.app j",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Preadditive C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nt : BinaryBicone X Y\nht : IsLimit (BinaryBicone.toCone t)\n⊢ t.inr = IsLimit.lift ht (BinaryFan.mk 0 (𝟙 Y))",
"tactic": "apply ht.uniq (BinaryFan.mk 0 (𝟙 Y))"
},
{
"state_after": "no goals",
"state_before": "case x\nC : Type u\ninst✝² : Category C\ninst✝¹ : Preadditive C\nJ : Type\ninst✝ : Fintype J\nX Y : C\nt : BinaryBicone X Y\nht : IsLimit (BinaryBicone.toCone t)\n⊢ ∀ (j : Discrete WalkingPair), t.inr ≫ (BinaryBicone.toCone t).π.app j = (BinaryFan.mk 0 (𝟙 Y)).π.app j",
"tactic": "rintro ⟨⟨⟩⟩ <;> dsimp <;> simp"
}
] | [
357,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
355,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean | map_mul_right_nhds_one | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : TopologicalSpace α\nf : α → G\ns : Set α\nx✝ : α\nx : G\n⊢ map (fun y => y * x) (𝓝 1) = 𝓝 x",
"tactic": "simp"
}
] | [
837,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
837,
1
] |
Mathlib/Logic/Function/Basic.lean | Function.extend_injective | [
{
"state_after": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\n⊢ g₁ = g₂",
"state_before": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\n⊢ Injective fun g => extend f g e'",
"tactic": "intro g₁ g₂ hg"
},
{
"state_after": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\n⊢ g₁ x = g₂ x",
"state_before": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\n⊢ g₁ = g₂",
"tactic": "refine' funext fun x ↦ _"
},
{
"state_after": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\nH : (fun g => extend f g e') g₁ (f x) = (fun g => extend f g e') g₂ (f x)\n⊢ g₁ x = g₂ x",
"state_before": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\n⊢ g₁ x = g₂ x",
"tactic": "have H := congr_fun hg (f x)"
},
{
"state_after": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\nH : g₁ x = g₂ x\n⊢ g₁ x = g₂ x",
"state_before": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\nH : (fun g => extend f g e') g₁ (f x) = (fun g => extend f g e') g₂ (f x)\n⊢ g₁ x = g₂ x",
"tactic": "simp only [hf.extend_apply] at H"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\nhf : Injective f\ne' : β → γ\ng₁ g₂ : α → γ\nhg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂\nx : α\nH : g₁ x = g₂ x\n⊢ g₁ x = g₂ x",
"tactic": "exact H"
}
] | [
771,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
766,
1
] |
Mathlib/Data/Set/Basic.lean | Set.union_subset_union_left | [] | [
838,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
837,
1
] |
Mathlib/Data/Set/Image.lean | Set.preimage_id' | [] | [
130,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearEquiv.range | [] | [
2135,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2134,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.sin_coe_pi | [
{
"state_after": "no goals",
"state_before": "⊢ sin ↑π = 0",
"tactic": "rw [sin_coe, Real.sin_pi]"
}
] | [
362,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
362,
1
] |
Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.closure_mono | [] | [
293,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
292,
1
] |
Mathlib/RingTheory/Polynomial/Pochhammer.lean | Polynomial.mul_X_add_nat_cast_comp | [
{
"state_after": "no goals",
"state_before": "S : Type u\ninst✝ : Semiring S\np q : S[X]\nn : ℕ\n⊢ comp (p * (X + ↑n)) q = comp p q * (q + ↑n)",
"tactic": "rw [mul_add, add_comp, mul_X_comp, ← Nat.cast_comm, nat_cast_mul_comp, Nat.cast_comm, mul_add]"
}
] | [
134,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
132,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.mem_iInf_of_mem | [] | [
586,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
585,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.forall_measure_inter_spanningSets_eq_zero | [
{
"state_after": "α : Type u_1\nβ : Type ?u.732836\nγ : Type ?u.732839\nδ : Type ?u.732842\nι : Type ?u.732845\nR : Type ?u.732848\nR' : Type ?u.732851\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\n⊢ (∀ (n : ℕ), ↑↑μ (s ∩ spanningSets μ n) = 0) ↔ ↑↑μ (⋃ (n : ℕ), s ∩ spanningSets μ n) = 0",
"state_before": "α : Type u_1\nβ : Type ?u.732836\nγ : Type ?u.732839\nδ : Type ?u.732842\nι : Type ?u.732845\nR : Type ?u.732848\nR' : Type ?u.732851\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\n⊢ (∀ (n : ℕ), ↑↑μ (s ∩ spanningSets μ n) = 0) ↔ ↑↑μ s = 0",
"tactic": "nth_rw 2 [show s = ⋃ n, s ∩ spanningSets μ n by\n rw [← inter_iUnion, iUnion_spanningSets, inter_univ] ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.732836\nγ : Type ?u.732839\nδ : Type ?u.732842\nι : Type ?u.732845\nR : Type ?u.732848\nR' : Type ?u.732851\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\n⊢ (∀ (n : ℕ), ↑↑μ (s ∩ spanningSets μ n) = 0) ↔ ↑↑μ (⋃ (n : ℕ), s ∩ spanningSets μ n) = 0",
"tactic": "rw [measure_iUnion_null_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.732836\nγ : Type ?u.732839\nδ : Type ?u.732842\nι : Type ?u.732845\nR : Type ?u.732848\nR' : Type ?u.732851\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\n⊢ s = ⋃ (n : ℕ), s ∩ spanningSets μ n",
"tactic": "rw [← inter_iUnion, iUnion_spanningSets, inter_univ]"
}
] | [
3560,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3556,
1
] |
Mathlib/Data/Complex/Exponential.lean | Real.cosh_neg | [
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ ↑(cosh (-x)) = ↑(cosh x)",
"tactic": "simp"
}
] | [
1370,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1369,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.