file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.iUnion_eq_empty
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\n⊢ Prepartition.iUnion π₁ = ∅ ↔ π₁ = ⊥",
"tactic": "simp [← injective_boxes.eq_iff, Finset.ext_iff, Prepartition.iUnion, imp_false]"
}
] |
[
239,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.image_val_of_injOn
|
[] |
[
408,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.smul_finset_neg
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.937268\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.937277\ninst✝³ : Monoid α\ninst✝² : AddGroup β\ninst✝¹ : DistribMulAction α β\ninst✝ : DecidableEq β\na : α\ns : Finset α\nt : Finset β\n⊢ a • -t = -(a • t)",
"tactic": "simp only [← image_smul, ← image_neg, Function.comp, image_image, smul_neg]"
}
] |
[
2145,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2144,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_mulSupport
|
[] |
[
178,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.nfpBFamily_eq_self
|
[] |
[
340,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/LinearAlgebra/Matrix/Basis.lean
|
Basis.toMatrix_reindex'
|
[
{
"state_after": "case a.h\nι : Type u_1\nι' : Type u_2\nκ : Type ?u.604344\nκ' : Type ?u.604347\nR : Type u_3\nM : Type u_4\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type ?u.604540\nM₂ : Type ?u.604543\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne✝ : Basis ι R M\nv✝ : ι' → M\ni : ι\nj : ι'\nN : Type ?u.605288\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Fintype κ\ninst✝³ : Fintype κ'\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\nb : Basis ι R M\nv : ι' → M\ne : ι ≃ ι'\ni✝ x✝ : ι'\n⊢ toMatrix (reindex b e) v i✝ x✝ = ↑(reindexAlgEquiv R e) (toMatrix b (v ∘ ↑e)) i✝ x✝",
"state_before": "ι : Type u_1\nι' : Type u_2\nκ : Type ?u.604344\nκ' : Type ?u.604347\nR : Type u_3\nM : Type u_4\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type ?u.604540\nM₂ : Type ?u.604543\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne✝ : Basis ι R M\nv✝ : ι' → M\ni : ι\nj : ι'\nN : Type ?u.605288\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Fintype κ\ninst✝³ : Fintype κ'\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\nb : Basis ι R M\nv : ι' → M\ne : ι ≃ ι'\n⊢ toMatrix (reindex b e) v = ↑(reindexAlgEquiv R e) (toMatrix b (v ∘ ↑e))",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nι : Type u_1\nι' : Type u_2\nκ : Type ?u.604344\nκ' : Type ?u.604347\nR : Type u_3\nM : Type u_4\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type ?u.604540\nM₂ : Type ?u.604543\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne✝ : Basis ι R M\nv✝ : ι' → M\ni : ι\nj : ι'\nN : Type ?u.605288\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb✝ : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Fintype κ\ninst✝³ : Fintype κ'\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\nb : Basis ι R M\nv : ι' → M\ne : ι ≃ ι'\ni✝ x✝ : ι'\n⊢ toMatrix (reindex b e) v i✝ x✝ = ↑(reindexAlgEquiv R e) (toMatrix b (v ∘ ↑e)) i✝ x✝",
"tactic": "simp only [Basis.toMatrix_apply, Basis.repr_reindex, Matrix.reindexAlgEquiv_apply,\n Matrix.reindex_apply, Matrix.submatrix_apply, Function.comp_apply, e.apply_symm_apply,\n Finsupp.mapDomain_equiv_apply]"
}
] |
[
239,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.comp_smulₛₗ
|
[
{
"state_after": "case h\nR : Type u_5\nR₂ : Type u_1\nR₃ : Type u_3\nS : Type ?u.1057088\nS₃ : Type ?u.1057091\ninst✝³⁶ : Semiring R\ninst✝³⁵ : Semiring R₂\ninst✝³⁴ : Semiring R₃\ninst✝³³ : Monoid S\ninst✝³² : Monoid S₃\nM : Type u_6\ninst✝³¹ : TopologicalSpace M\ninst✝³⁰ : AddCommMonoid M\ninst✝²⁹ : Module R M\nM₂ : Type u_2\ninst✝²⁸ : TopologicalSpace M₂\ninst✝²⁷ : AddCommMonoid M₂\ninst✝²⁶ : Module R₂ M₂\nM₃ : Type u_4\ninst✝²⁵ : TopologicalSpace M₃\ninst✝²⁴ : AddCommMonoid M₃\ninst✝²³ : Module R₃ M₃\nN₂ : Type ?u.1057214\ninst✝²² : TopologicalSpace N₂\ninst✝²¹ : AddCommMonoid N₂\ninst✝²⁰ : Module R N₂\nN₃ : Type ?u.1057247\ninst✝¹⁹ : TopologicalSpace N₃\ninst✝¹⁸ : AddCommMonoid N₃\ninst✝¹⁷ : Module R N₃\ninst✝¹⁶ : DistribMulAction S₃ M₃\ninst✝¹⁵ : SMulCommClass R₃ S₃ M₃\ninst✝¹⁴ : ContinuousConstSMul S₃ M₃\ninst✝¹³ : DistribMulAction S N₃\ninst✝¹² : SMulCommClass R S N₃\ninst✝¹¹ : ContinuousConstSMul S N₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹⁰ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁹ : DistribMulAction S₃ M₂\ninst✝⁸ : ContinuousConstSMul S₃ M₂\ninst✝⁷ : SMulCommClass R₂ S₃ M₂\ninst✝⁶ : DistribMulAction S N₂\ninst✝⁵ : ContinuousConstSMul S N₂\ninst✝⁴ : SMulCommClass R S N₂\ninst✝³ : SMulCommClass R₂ R₂ M₂\ninst✝² : SMulCommClass R₃ R₃ M₃\ninst✝¹ : ContinuousConstSMul R₂ M₂\ninst✝ : ContinuousConstSMul R₃ M₃\nh : M₂ →SL[σ₂₃] M₃\nc : R₂\nf : M →SL[σ₁₂] M₂\nx : M\n⊢ ↑(comp h (c • f)) x = ↑(↑σ₂₃ c • comp h f) x",
"state_before": "R : Type u_5\nR₂ : Type u_1\nR₃ : Type u_3\nS : Type ?u.1057088\nS₃ : Type ?u.1057091\ninst✝³⁶ : Semiring R\ninst✝³⁵ : Semiring R₂\ninst✝³⁴ : Semiring R₃\ninst✝³³ : Monoid S\ninst✝³² : Monoid S₃\nM : Type u_6\ninst✝³¹ : TopologicalSpace M\ninst✝³⁰ : AddCommMonoid M\ninst✝²⁹ : Module R M\nM₂ : Type u_2\ninst✝²⁸ : TopologicalSpace M₂\ninst✝²⁷ : AddCommMonoid M₂\ninst✝²⁶ : Module R₂ M₂\nM₃ : Type u_4\ninst✝²⁵ : TopologicalSpace M₃\ninst✝²⁴ : AddCommMonoid M₃\ninst✝²³ : Module R₃ M₃\nN₂ : Type ?u.1057214\ninst✝²² : TopologicalSpace N₂\ninst✝²¹ : AddCommMonoid N₂\ninst✝²⁰ : Module R N₂\nN₃ : Type ?u.1057247\ninst✝¹⁹ : TopologicalSpace N₃\ninst✝¹⁸ : AddCommMonoid N₃\ninst✝¹⁷ : Module R N₃\ninst✝¹⁶ : DistribMulAction S₃ M₃\ninst✝¹⁵ : SMulCommClass R₃ S₃ M₃\ninst✝¹⁴ : ContinuousConstSMul S₃ M₃\ninst✝¹³ : DistribMulAction S N₃\ninst✝¹² : SMulCommClass R S N₃\ninst✝¹¹ : ContinuousConstSMul S N₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹⁰ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁹ : DistribMulAction S₃ M₂\ninst✝⁸ : ContinuousConstSMul S₃ M₂\ninst✝⁷ : SMulCommClass R₂ S₃ M₂\ninst✝⁶ : DistribMulAction S N₂\ninst✝⁵ : ContinuousConstSMul S N₂\ninst✝⁴ : SMulCommClass R S N₂\ninst✝³ : SMulCommClass R₂ R₂ M₂\ninst✝² : SMulCommClass R₃ R₃ M₃\ninst✝¹ : ContinuousConstSMul R₂ M₂\ninst✝ : ContinuousConstSMul R₃ M₃\nh : M₂ →SL[σ₂₃] M₃\nc : R₂\nf : M →SL[σ₁₂] M₂\n⊢ comp h (c • f) = ↑σ₂₃ c • comp h f",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_5\nR₂ : Type u_1\nR₃ : Type u_3\nS : Type ?u.1057088\nS₃ : Type ?u.1057091\ninst✝³⁶ : Semiring R\ninst✝³⁵ : Semiring R₂\ninst✝³⁴ : Semiring R₃\ninst✝³³ : Monoid S\ninst✝³² : Monoid S₃\nM : Type u_6\ninst✝³¹ : TopologicalSpace M\ninst✝³⁰ : AddCommMonoid M\ninst✝²⁹ : Module R M\nM₂ : Type u_2\ninst✝²⁸ : TopologicalSpace M₂\ninst✝²⁷ : AddCommMonoid M₂\ninst✝²⁶ : Module R₂ M₂\nM₃ : Type u_4\ninst✝²⁵ : TopologicalSpace M₃\ninst✝²⁴ : AddCommMonoid M₃\ninst✝²³ : Module R₃ M₃\nN₂ : Type ?u.1057214\ninst✝²² : TopologicalSpace N₂\ninst✝²¹ : AddCommMonoid N₂\ninst✝²⁰ : Module R N₂\nN₃ : Type ?u.1057247\ninst✝¹⁹ : TopologicalSpace N₃\ninst✝¹⁸ : AddCommMonoid N₃\ninst✝¹⁷ : Module R N₃\ninst✝¹⁶ : DistribMulAction S₃ M₃\ninst✝¹⁵ : SMulCommClass R₃ S₃ M₃\ninst✝¹⁴ : ContinuousConstSMul S₃ M₃\ninst✝¹³ : DistribMulAction S N₃\ninst✝¹² : SMulCommClass R S N₃\ninst✝¹¹ : ContinuousConstSMul S N₃\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\ninst✝¹⁰ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁹ : DistribMulAction S₃ M₂\ninst✝⁸ : ContinuousConstSMul S₃ M₂\ninst✝⁷ : SMulCommClass R₂ S₃ M₂\ninst✝⁶ : DistribMulAction S N₂\ninst✝⁵ : ContinuousConstSMul S N₂\ninst✝⁴ : SMulCommClass R S N₂\ninst✝³ : SMulCommClass R₂ R₂ M₂\ninst✝² : SMulCommClass R₃ R₃ M₃\ninst✝¹ : ContinuousConstSMul R₂ M₂\ninst✝ : ContinuousConstSMul R₃ M₃\nh : M₂ →SL[σ₂₃] M₃\nc : R₂\nf : M →SL[σ₁₂] M₂\nx : M\n⊢ ↑(comp h (c • f)) x = ↑(↑σ₂₃ c • comp h f) x",
"tactic": "simp only [coe_smul', coe_comp', Function.comp_apply, Pi.smul_apply,\n ContinuousLinearMap.map_smulₛₗ]"
}
] |
[
1530,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1525,
1
] |
Mathlib/Data/Nat/Fib.lean
|
Nat.fib_two_mul
|
[
{
"state_after": "case zero\n\n⊢ fib (2 * zero) = fib zero * (2 * fib (zero + 1) - fib zero)\n\ncase succ\nn✝ : ℕ\n⊢ fib (2 * succ n✝) = fib (succ n✝) * (2 * fib (succ n✝ + 1) - fib (succ n✝))",
"state_before": "n : ℕ\n⊢ fib (2 * n) = fib n * (2 * fib (n + 1) - fib n)",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ fib (2 * zero) = fib zero * (2 * fib (zero + 1) - fib zero)",
"tactic": "simp"
},
{
"state_after": "case succ\nn✝ : ℕ\n⊢ fib (n✝ + 1) * fib n✝ + (fib n✝ + fib (n✝ + 1)) * fib (n✝ + 1) =\n fib (n✝ + 1) * (fib n✝ + fib (n✝ + 1) + (fib n✝ + fib (n✝ + 1)) - fib (n✝ + 1))",
"state_before": "case succ\nn✝ : ℕ\n⊢ fib (2 * succ n✝) = fib (succ n✝) * (2 * fib (succ n✝ + 1) - fib (succ n✝))",
"tactic": "rw [Nat.succ_eq_add_one, two_mul, ← add_assoc, fib_add, fib_add_two, two_mul]"
},
{
"state_after": "case succ\nn✝ : ℕ\n⊢ fib (n✝ + 1) * fib n✝ + (fib n✝ + fib (n✝ + 1)) * fib (n✝ + 1) = fib (n✝ + 1) * (fib n✝ + fib (n✝ + 1) + fib n✝)",
"state_before": "case succ\nn✝ : ℕ\n⊢ fib (n✝ + 1) * fib n✝ + (fib n✝ + fib (n✝ + 1)) * fib (n✝ + 1) =\n fib (n✝ + 1) * (fib n✝ + fib (n✝ + 1) + (fib n✝ + fib (n✝ + 1)) - fib (n✝ + 1))",
"tactic": "simp only [← add_assoc, add_tsub_cancel_right]"
},
{
"state_after": "no goals",
"state_before": "case succ\nn✝ : ℕ\n⊢ fib (n✝ + 1) * fib n✝ + (fib n✝ + fib (n✝ + 1)) * fib (n✝ + 1) = fib (n✝ + 1) * (fib n✝ + fib (n✝ + 1) + fib n✝)",
"tactic": "ring"
}
] |
[
165,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.isComplement'_iff_card_mul_and_disjoint
|
[] |
[
544,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
Mathlib/Order/Lattice.lean
|
Antitone.max
|
[] |
[
1178,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1175,
11
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.Monic.geom_sum
|
[
{
"state_after": "R : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\nn : ℕ\nhn : n ≠ 0\n✝ : Nontrivial R\n⊢ Monic (∑ i in range n, P ^ i)",
"state_before": "R : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\nn : ℕ\nhn : n ≠ 0\n⊢ Monic (∑ i in range n, P ^ i)",
"tactic": "nontriviality R"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ Monic (∑ i in range (Nat.succ n), P ^ i)",
"state_before": "R : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\nn : ℕ\nhn : n ≠ 0\n✝ : Nontrivial R\n⊢ Monic (∑ i in range n, P ^ i)",
"tactic": "obtain ⟨n, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ Monic (P ^ n + ∑ i in range n, P ^ i)",
"state_before": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ Monic (∑ i in range (Nat.succ n), P ^ i)",
"tactic": "rw [geom_sum_succ']"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ degree (∑ i in range n, P ^ i) < degree (P ^ n)",
"state_before": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ Monic (P ^ n + ∑ i in range n, P ^ i)",
"tactic": "refine' (hP.pow _).add_of_left _"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ (sup (range n) fun b => degree (P ^ b)) < degree (P ^ n)",
"state_before": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ degree (∑ i in range n, P ^ i) < degree (P ^ n)",
"tactic": "refine' lt_of_le_of_lt (degree_sum_le _ _) _"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ ∀ (b : ℕ), b ∈ range n → degree (P ^ b) < degree (P ^ n)\n\ncase intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ ⊥ < degree (P ^ n)",
"state_before": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ (sup (range n) fun b => degree (P ^ b)) < degree (P ^ n)",
"tactic": "rw [Finset.sup_lt_iff]"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ ∀ (b : ℕ), b < n → ↑(natDegree (P ^ b)) < ↑(natDegree (P ^ n))",
"state_before": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ ∀ (b : ℕ), b ∈ range n → degree (P ^ b) < degree (P ^ n)",
"tactic": "simp only [Finset.mem_range, degree_eq_natDegree (hP.pow _).ne_zero]"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ ∀ (b : ℕ), b < n → b * natDegree P < n * natDegree P",
"state_before": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ ∀ (b : ℕ), b < n → ↑(natDegree (P ^ b)) < ↑(natDegree (P ^ n))",
"tactic": "simp only [Nat.cast_withBot, WithBot.coe_lt_coe, hP.natDegree_pow]"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\nk : ℕ\n⊢ k < n → k * natDegree P < n * natDegree P",
"state_before": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ ∀ (b : ℕ), b < n → b * natDegree P < n * natDegree P",
"tactic": "intro k"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\nk : ℕ\n⊢ k < n → k * natDegree P < n * natDegree P",
"tactic": "exact nsmul_lt_nsmul hdeg"
},
{
"state_after": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ ¬P ^ n = 0",
"state_before": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ ⊥ < degree (P ^ n)",
"tactic": "rw [bot_lt_iff_ne_bot, Ne.def, degree_eq_bot]"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\nS : Type ?u.54680\ninst✝ : Semiring R\nP : R[X]\nhP : Monic P\nhdeg : 0 < natDegree P\n✝ : Nontrivial R\nn : ℕ\nhn : Nat.succ n ≠ 0\n⊢ ¬P ^ n = 0",
"tactic": "exact (hP.pow _).ne_zero"
}
] |
[
244,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.cauchy_add
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\na b : Cauchy abs\n⊢ (Real.add { cauchy := a } { cauchy := b }).cauchy = { cauchy := a }.cauchy + { cauchy := b }.cauchy",
"tactic": "rw [add_def]"
}
] |
[
153,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.map₂_div
|
[] |
[
435,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
Finset.sum_apply'
|
[] |
[
626,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
625,
1
] |
Mathlib/Order/SupIndep.lean
|
CompleteLattice.Independent.injective
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\n⊢ i = j",
"state_before": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\n⊢ Injective t",
"tactic": "intro i j h"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\n⊢ False",
"state_before": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\n⊢ i = j",
"tactic": "by_contra' contra"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\n⊢ t j = ⊥",
"state_before": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\n⊢ False",
"tactic": "apply h_ne_bot j"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\n⊢ t j ≤ ⨆ (k : ι) (_ : k ≠ i), t k",
"state_before": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\n⊢ t j = ⊥",
"tactic": "suffices t j ≤ ⨆ (k) (_ : k ≠ i), t k by\n replace ht := (ht i).mono_right this\n rwa [h, disjoint_self] at ht"
},
{
"state_after": "case contra\nα : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\n⊢ j ≠ i\n\nα : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : j ≠ i\n⊢ t j ≤ ⨆ (k : ι) (_ : k ≠ i), t k",
"state_before": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\n⊢ t j ≤ ⨆ (k : ι) (_ : k ≠ i), t k",
"tactic": "replace contra : j ≠ i"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : j ≠ i\n⊢ t j ≤ ⨆ (k : ι) (_ : k ≠ i), t k",
"tactic": "exact @le_iSup₂ _ _ _ _ (fun x _ => t x) j contra"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\nthis : t j ≤ ⨆ (k : ι) (_ : k ≠ i), t k\nht : Disjoint (t i) (t j)\n⊢ t j = ⊥",
"state_before": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\nthis : t j ≤ ⨆ (k : ι) (_ : k ≠ i), t k\n⊢ t j = ⊥",
"tactic": "replace ht := (ht i).mono_right this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\nthis : t j ≤ ⨆ (k : ι) (_ : k ≠ i), t k\nht : Disjoint (t i) (t j)\n⊢ t j = ⊥",
"tactic": "rwa [h, disjoint_self] at ht"
},
{
"state_after": "no goals",
"state_before": "case contra\nα : Type u_2\nβ : Type ?u.53351\nι : Type u_1\nι' : Type ?u.53357\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht✝ ht : Independent t\nh_ne_bot : ∀ (i : ι), t i ≠ ⊥\ni j : ι\nh : t i = t j\ncontra : i ≠ j\n⊢ j ≠ i",
"tactic": "exact Ne.symm contra"
}
] |
[
322,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
Class.sUnion_empty
|
[
{
"state_after": "case a\nz✝ : ZFSet\n⊢ (⋃₀ ∅) z✝ ↔ ∅ z✝",
"state_before": "⊢ ⋃₀ ∅ = ∅",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nz✝ : ZFSet\n⊢ (⋃₀ ∅) z✝ ↔ ∅ z✝",
"tactic": "simp"
}
] |
[
1728,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1726,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
Filter.Tendsto.cauchySeq
|
[] |
[
190,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.natTrailingDegree_monomial_le
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nha : a = 0\n⊢ natTrailingDegree (↑(monomial n) a) ≤ n",
"tactic": "simp [ha]"
}
] |
[
213,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Order/CompleteLattice.lean
|
Prod.iSup_mk
|
[] |
[
1888,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1886,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sqrtTwoAddSeries_lt_two
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ sqrtTwoAddSeries 0 0 < 2",
"tactic": "norm_num"
},
{
"state_after": "x : ℝ\nn : ℕ\n⊢ sqrtTwoAddSeries 0 (n + 1) < sqrt (2 ^ 2)",
"state_before": "x : ℝ\nn : ℕ\n⊢ sqrtTwoAddSeries 0 (n + 1) < 2",
"tactic": "refine' lt_of_lt_of_le _ (sqrt_sq zero_lt_two.le).le"
},
{
"state_after": "x : ℝ\nn : ℕ\n⊢ sqrtTwoAddSeries 0 n < 2 ^ 2 - 2\n\nx : ℝ\nn : ℕ\n⊢ 0 ≤ 2 + sqrtTwoAddSeries 0 n",
"state_before": "x : ℝ\nn : ℕ\n⊢ sqrtTwoAddSeries 0 (n + 1) < sqrt (2 ^ 2)",
"tactic": "rw [sqrtTwoAddSeries, sqrt_lt_sqrt_iff, ← lt_sub_iff_add_lt']"
},
{
"state_after": "x : ℝ\nn : ℕ\n⊢ 2 ≤ 2 ^ 2 - 2",
"state_before": "x : ℝ\nn : ℕ\n⊢ sqrtTwoAddSeries 0 n < 2 ^ 2 - 2",
"tactic": "refine' (sqrtTwoAddSeries_lt_two n).trans_le _"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nn : ℕ\n⊢ 2 ≤ 2 ^ 2 - 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nn : ℕ\n⊢ 0 ≤ 2 + sqrtTwoAddSeries 0 n",
"tactic": "exact add_nonneg zero_le_two (sqrtTwoAddSeries_zero_nonneg n)"
}
] |
[
708,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
701,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.coeff_map
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\n⊢ ∑ n_1 in support p, coeff (↑(RingHom.comp C f) (coeff p n_1) * X ^ n_1) n = ↑f (coeff p n)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\n⊢ coeff (map f p) n = ↑f (coeff p n)",
"tactic": "rw [map, eval₂_def, coeff_sum, sum]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\n⊢ ∑ n_1 in support p, coeff (↑(RingHom.comp C f) (coeff p n_1) * X ^ n_1) n =\n ∑ x in support p, ↑f (coeff (↑C (coeff p x) * X ^ x) n)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\n⊢ ∑ n_1 in support p, coeff (↑(RingHom.comp C f) (coeff p n_1) * X ^ n_1) n = ↑f (coeff p n)",
"tactic": "conv_rhs => rw [← sum_C_mul_X_pow_eq p, coeff_sum, sum, map_sum]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nn x : ℕ\n_hx : x ∈ support p\n⊢ coeff (↑(RingHom.comp C f) (coeff p x) * X ^ x) n = ↑f (coeff (↑C (coeff p x) * X ^ x) n)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\n⊢ ∑ n_1 in support p, coeff (↑(RingHom.comp C f) (coeff p n_1) * X ^ n_1) n =\n ∑ x in support p, ↑f (coeff (↑C (coeff p x) * X ^ x) n)",
"tactic": "refine' Finset.sum_congr rfl fun x _hx => _"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nn x : ℕ\n_hx : x ∈ support p\n⊢ (if n = x then ↑f (coeff p x) else 0) = ↑f (if n = x then coeff p x else 0)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nn x : ℕ\n_hx : x ∈ support p\n⊢ coeff (↑(RingHom.comp C f) (coeff p x) * X ^ x) n = ↑f (coeff (↑C (coeff p x) * X ^ x) n)",
"tactic": "simp [Function.comp, coeff_C_mul_X_pow, - map_mul, - coeff_C_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n✝ : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nn x : ℕ\n_hx : x ∈ support p\n⊢ (if n = x then ↑f (coeff p x) else 0) = ↑f (if n = x then coeff p x else 0)",
"tactic": "split_ifs <;> simp [f.map_zero]"
}
] |
[
793,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
787,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
Int.norm_eq_abs
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.1111123\n𝕜 : Type ?u.1111126\nα : Type ?u.1111129\nι : Type ?u.1111132\nκ : Type ?u.1111135\nE : Type ?u.1111138\nF : Type ?u.1111141\nG : Type ?u.1111144\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nn : ℤ\n⊢ ‖↑n‖ = ↑(abs n)",
"tactic": "rw [Real.norm_eq_abs, cast_abs]"
}
] |
[
1791,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1790,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
nsmul_zero
|
[
{
"state_after": "case zero\nα : Type ?u.8662\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\n⊢ Nat.zero • 0 = 0\n\ncase succ\nα : Type ?u.8662\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\nn : ℕ\nih : n • 0 = 0\n⊢ Nat.succ n • 0 = 0",
"state_before": "α : Type ?u.8662\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\nn : ℕ\n⊢ n • 0 = 0",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type ?u.8662\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\n⊢ Nat.zero • 0 = 0",
"tactic": "exact zero_nsmul _"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type ?u.8662\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\nn : ℕ\nih : n • 0 = 0\n⊢ Nat.succ n • 0 = 0",
"tactic": "rw [succ_nsmul, ih, add_zero]"
}
] |
[
78,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Std/Classes/LawfulMonad.lean
|
SatisfiesM.pure
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα✝ : Type u_1\np : α✝ → Prop\na : α✝\ninst✝¹ : Applicative m\ninst✝ : LawfulApplicative m\nh : p a\n⊢ Subtype.val <$> pure { val := a, property := h } = pure a",
"tactic": "simp"
}
] |
[
121,
73
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
120,
11
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm_smul_le_snorm_top_mul_snorm
|
[] |
[
1503,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1500,
1
] |
Mathlib/Control/Traversable/Equiv.lean
|
Equiv.comp_traverse
|
[
{
"state_after": "t t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β γ : Type u\nf : β → F γ\ng : α → G β\nx : t' α\n⊢ Comp.mk (((fun x => ↑(eqv γ) <$> x) ∘ traverse f) <$> traverse g (↑(eqv α).symm x)) =\n Comp.mk (((fun x => ↑(eqv γ) <$> traverse f (↑(eqv β).symm x)) ∘ ↑(eqv β)) <$> traverse g (↑(eqv α).symm x))",
"state_before": "t t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β γ : Type u\nf : β → F γ\ng : α → G β\nx : t' α\n⊢ Equiv.traverse eqv (Comp.mk ∘ map f ∘ g) x = Comp.mk (Equiv.traverse eqv f <$> Equiv.traverse eqv g x)",
"tactic": "simp [Equiv.traverse, comp_traverse, functor_norm]"
},
{
"state_after": "case e_x.e_a\nt t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β γ : Type u\nf : β → F γ\ng : α → G β\nx : t' α\n⊢ (fun x => ↑(eqv γ) <$> x) ∘ traverse f = (fun x => ↑(eqv γ) <$> traverse f (↑(eqv β).symm x)) ∘ ↑(eqv β)",
"state_before": "t t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β γ : Type u\nf : β → F γ\ng : α → G β\nx : t' α\n⊢ Comp.mk (((fun x => ↑(eqv γ) <$> x) ∘ traverse f) <$> traverse g (↑(eqv α).symm x)) =\n Comp.mk (((fun x => ↑(eqv γ) <$> traverse f (↑(eqv β).symm x)) ∘ ↑(eqv β)) <$> traverse g (↑(eqv α).symm x))",
"tactic": "congr"
},
{
"state_after": "case e_x.e_a.h\nt t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β γ : Type u\nf : β → F γ\ng : α → G β\nx : t' α\nx✝ : t β\n⊢ ((fun x => ↑(eqv γ) <$> x) ∘ traverse f) x✝ = ((fun x => ↑(eqv γ) <$> traverse f (↑(eqv β).symm x)) ∘ ↑(eqv β)) x✝",
"state_before": "case e_x.e_a\nt t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β γ : Type u\nf : β → F γ\ng : α → G β\nx : t' α\n⊢ (fun x => ↑(eqv γ) <$> x) ∘ traverse f = (fun x => ↑(eqv γ) <$> traverse f (↑(eqv β).symm x)) ∘ ↑(eqv β)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case e_x.e_a.h\nt t' : Type u → Type u\neqv : (α : Type u) → t α ≃ t' α\ninst✝⁵ : Traversable t\ninst✝⁴ : IsLawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β γ : Type u\nf : β → F γ\ng : α → G β\nx : t' α\nx✝ : t β\n⊢ ((fun x => ↑(eqv γ) <$> x) ∘ traverse f) x✝ = ((fun x => ↑(eqv γ) <$> traverse f (↑(eqv β).symm x)) ∘ ↑(eqv β)) x✝",
"tactic": "simp"
}
] |
[
150,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
11
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullbackConeOfLeftIso_x
|
[] |
[
1617,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1617,
1
] |
Mathlib/GroupTheory/FreeProduct.lean
|
FreeProduct.NeWord.inv_head
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nM : ι → Type ?u.649174\ninst✝² : (i : ι) → Monoid (M i)\nN : Type ?u.649185\ninst✝¹ : Monoid N\nG : ι → Type u_2\ninst✝ : (i : ι) → Group (G i)\ni j : ι\nw : NeWord G i j\n⊢ head (inv w) = (last w)⁻¹",
"tactic": "induction w <;> simp [inv, *]"
}
] |
[
685,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
684,
1
] |
Mathlib/CategoryTheory/Endofunctor/Algebra.lean
|
CategoryTheory.Endofunctor.Coalgebra.iso_of_iso
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nF : C ⥤ C\nV V₀ V₁ V₂ : Coalgebra F\nf✝ : V₀ ⟶ V₁\ng : V₁ ⟶ V₂\nf : V₀ ⟶ V₁\ninst✝ : IsIso f.f\n⊢ V₀.str ≫ F.map f.f ≫ F.map (inv f.f) = V₀.str",
"state_before": "C : Type u\ninst✝¹ : Category C\nF : C ⥤ C\nV V₀ V₁ V₂ : Coalgebra F\nf✝ : V₀ ⟶ V₁\ng : V₁ ⟶ V₂\nf : V₀ ⟶ V₁\ninst✝ : IsIso f.f\n⊢ V₁.str ≫ F.map (inv f.f) = inv f.f ≫ V₀.str",
"tactic": "rw [IsIso.eq_inv_comp f.1, ← Category.assoc, ← f.h, Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nF : C ⥤ C\nV V₀ V₁ V₂ : Coalgebra F\nf✝ : V₀ ⟶ V₁\ng : V₁ ⟶ V₂\nf : V₀ ⟶ V₁\ninst✝ : IsIso f.f\n⊢ V₀.str ≫ F.map f.f ≫ F.map (inv f.f) = V₀.str",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nF : C ⥤ C\nV V₀ V₁ V₂ : Coalgebra F\nf✝ : V₀ ⟶ V₁\ng : V₁ ⟶ V₂\nf : V₀ ⟶ V₁\ninst✝ : IsIso f.f\n⊢ f ≫ Hom.mk (inv f.f) = 𝟙 V₀",
"tactic": "aesop_cat"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nF : C ⥤ C\nV V₀ V₁ V₂ : Coalgebra F\nf✝ : V₀ ⟶ V₁\ng : V₁ ⟶ V₂\nf : V₀ ⟶ V₁\ninst✝ : IsIso f.f\n⊢ Hom.mk (inv f.f) ≫ f = 𝟙 V₁",
"tactic": "aesop_cat"
}
] |
[
378,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean
|
Ideal.hasBasis_nhds_zero_adic
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\n⊢ U ∈ 𝓝 0 ↔ ∃ i, True ∧ ↑(I ^ i) ⊆ U",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\n⊢ ∀ (t : Set R), t ∈ 𝓝 0 ↔ ∃ i, True ∧ ↑(I ^ i) ⊆ t",
"tactic": "intro U"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\n⊢ (∃ i, i ∈ RingFilterBasis.toAddGroupFilterBasis ∧ id i ⊆ U) ↔ ∃ i, True ∧ ↑(I ^ i) ⊆ U",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\n⊢ U ∈ 𝓝 0 ↔ ∃ i, True ∧ ↑(I ^ i) ⊆ U",
"tactic": "rw [I.ringFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]"
},
{
"state_after": "case mp\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\n⊢ (∃ i, i ∈ RingFilterBasis.toAddGroupFilterBasis ∧ id i ⊆ U) → ∃ i, True ∧ ↑(I ^ i) ⊆ U\n\ncase mpr\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\n⊢ (∃ i, True ∧ ↑(I ^ i) ⊆ U) → ∃ i, i ∈ RingFilterBasis.toAddGroupFilterBasis ∧ id i ⊆ U",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\n⊢ (∃ i, i ∈ RingFilterBasis.toAddGroupFilterBasis ∧ id i ⊆ U) ↔ ∃ i, True ∧ ↑(I ^ i) ⊆ U",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : id ↑((fun i => toAddSubgroup (I ^ i • ⊤)) i) ⊆ U\n⊢ ∃ i, True ∧ ↑(I ^ i) ⊆ U",
"state_before": "case mp\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\n⊢ (∃ i, i ∈ RingFilterBasis.toAddGroupFilterBasis ∧ id i ⊆ U) → ∃ i, True ∧ ↑(I ^ i) ⊆ U",
"tactic": "rintro ⟨-, ⟨i, rfl⟩, h⟩"
},
{
"state_after": "case mp.intro.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : ↑(I ^ i) ⊆ U\n⊢ ∃ i, True ∧ ↑(I ^ i) ⊆ U",
"state_before": "case mp.intro.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : id ↑((fun i => toAddSubgroup (I ^ i • ⊤)) i) ⊆ U\n⊢ ∃ i, True ∧ ↑(I ^ i) ⊆ U",
"tactic": "replace h : ↑(I ^ i) ⊆ U := by simpa using h"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : ↑(I ^ i) ⊆ U\n⊢ ∃ i, True ∧ ↑(I ^ i) ⊆ U",
"tactic": "exact ⟨i, trivial, h⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : id ↑((fun i => toAddSubgroup (I ^ i • ⊤)) i) ⊆ U\n⊢ ↑(I ^ i) ⊆ U",
"tactic": "simpa using h"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : ↑(I ^ i) ⊆ U\n⊢ ∃ i, i ∈ RingFilterBasis.toAddGroupFilterBasis ∧ id i ⊆ U",
"state_before": "case mpr\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\n⊢ (∃ i, True ∧ ↑(I ^ i) ⊆ U) → ∃ i, i ∈ RingFilterBasis.toAddGroupFilterBasis ∧ id i ⊆ U",
"tactic": "rintro ⟨i, -, h⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : ↑(I ^ i) ⊆ U\n⊢ ∃ i, i ∈ RingFilterBasis.toAddGroupFilterBasis ∧ id i ⊆ U",
"tactic": "exact ⟨(I ^ i : Ideal R), ⟨i, by simp⟩, h⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nU : Set R\ni : ℕ\nh : ↑(I ^ i) ⊆ U\n⊢ ↑(I ^ i) = ↑((fun i => toAddSubgroup (I ^ i • ⊤)) i)",
"tactic": "simp"
}
] |
[
107,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
norm_le_mul_norm_add
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.86827\n𝕜 : Type ?u.86830\nα : Type ?u.86833\nι : Type ?u.86836\nκ : Type ?u.86839\nE : Type u_1\nF : Type ?u.86845\nG : Type ?u.86848\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nu v : E\n⊢ ‖u‖ = ‖u * v / v‖",
"tactic": "rw [mul_div_cancel'']"
}
] |
[
601,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
598,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.decode₂_ne_none_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7543\ninst✝ : Encodable α\nn : ℕ\n⊢ decode₂ α n ≠ none ↔ n ∈ Set.range encode",
"tactic": "simp_rw [Set.range, Set.mem_setOf_eq, Ne.def, Option.eq_none_iff_forall_not_mem,\n Encodable.mem_decode₂, not_forall, not_not]"
}
] |
[
217,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
|
MeasureTheory.tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable
|
[
{
"state_after": "α : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ : δ > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ δ",
"state_before": "α : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\n⊢ TendstoInMeasure μ f atTop g",
"tactic": "refine' fun ε hε => ENNReal.tendsto_atTop_zero.mpr fun δ hδ => _"
},
{
"state_after": "case pos\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ : δ > 0\nhδi : δ = ⊤\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ δ\n\ncase neg\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ : δ > 0\nhδi : ¬δ = ⊤\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ δ",
"state_before": "α : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ : δ > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ δ",
"tactic": "by_cases hδi : δ = ∞"
},
{
"state_after": "case neg.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : ↑δ > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"state_before": "case neg\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ : δ > 0\nhδi : ¬δ = ⊤\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ δ",
"tactic": "lift δ to ℝ≥0 using hδi"
},
{
"state_after": "case neg.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"state_before": "case neg.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : ↑δ > 0\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"tactic": "rw [gt_iff_lt, ENNReal.coe_pos, ← NNReal.coe_pos] at hδ"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ENNReal.ofReal ↑δ\nhunif : TendstoUniformlyOn (fun n => f n) g atTop (tᶜ)\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"state_before": "case neg.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"tactic": "obtain ⟨t, _, ht, hunif⟩ := tendstoUniformlyOn_of_ae_tendsto' hf hg hfg hδ"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : TendstoUniformlyOn (fun n => f n) g atTop (tᶜ)\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"state_before": "case neg.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ENNReal.ofReal ↑δ\nhunif : TendstoUniformlyOn (fun n => f n) g atTop (tᶜ)\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"tactic": "rw [ENNReal.ofReal_coe_nnreal] at ht"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"state_before": "case neg.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : TendstoUniformlyOn (fun n => f n) g atTop (tᶜ)\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"tactic": "rw [Metric.tendstoUniformlyOn_iff] at hunif"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"state_before": "case neg.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"tactic": "obtain ⟨N, hN⟩ := eventually_atTop.1 (hunif ε hε)"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\n⊢ ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"state_before": "case neg.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"tactic": "refine' ⟨N, fun n hn => _⟩"
},
{
"state_after": "case neg.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\nthis : {x | ε ≤ dist (f n x) (g x)} ⊆ t\n⊢ ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ\n\ncase this\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\n⊢ {x | ε ≤ dist (f n x) (g x)} ⊆ t",
"state_before": "case neg.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\n⊢ ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ",
"tactic": "suffices : { x : α | ε ≤ dist (f n x) (g x) } ⊆ t"
},
{
"state_after": "case this\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\n⊢ {x | ε ≤ dist (f n x) (g x)} ⊆ t",
"state_before": "case neg.intro.intro.intro.intro.intro\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\nthis : {x | ε ≤ dist (f n x) (g x)} ⊆ t\n⊢ ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ ↑δ\n\ncase this\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\n⊢ {x | ε ≤ dist (f n x) (g x)} ⊆ t",
"tactic": "exact (measure_mono this).trans ht"
},
{
"state_after": "case this\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\n⊢ tᶜ ⊆ {x | ε ≤ dist (f n x) (g x)}ᶜ",
"state_before": "case this\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\n⊢ {x | ε ≤ dist (f n x) (g x)} ⊆ t",
"tactic": "rw [← Set.compl_subset_compl]"
},
{
"state_after": "case this\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\nx : α\nhx : x ∈ tᶜ\n⊢ x ∈ {x | ε ≤ dist (f n x) (g x)}ᶜ",
"state_before": "case this\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\n⊢ tᶜ ⊆ {x | ε ≤ dist (f n x) (g x)}ᶜ",
"tactic": "intro x hx"
},
{
"state_after": "case this\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\nx : α\nhx : x ∈ tᶜ\n⊢ dist (g x) (f n x) < ε",
"state_before": "case this\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\nx : α\nhx : x ∈ tᶜ\n⊢ x ∈ {x | ε ≤ dist (f n x) (g x)}ᶜ",
"tactic": "rw [Set.mem_compl_iff, Set.nmem_setOf_iff, dist_comm, not_le]"
},
{
"state_after": "no goals",
"state_before": "case this\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0\nhδ : 0 < ↑δ\nt : Set α\nleft✝ : MeasurableSet t\nht : ↑↑μ t ≤ ↑δ\nhunif : ∀ (ε : ℝ), ε > 0 → ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), x ∈ tᶜ → dist (g x) (f n x) < ε\nN : ℕ\nhN : ∀ (b : ℕ), b ≥ N → ∀ (x : α), x ∈ tᶜ → dist (g x) (f b x) < ε\nn : ℕ\nhn : n ≥ N\nx : α\nhx : x ∈ tᶜ\n⊢ dist (g x) (f n x) < ε",
"tactic": "exact hN n hn x hx"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nι : Type ?u.3660\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : MetricSpace E\nf : ℕ → α → E\ng : α → E\ninst✝ : IsFiniteMeasure μ\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ : δ > 0\nhδi : δ = ⊤\n⊢ ∃ N, ∀ (n : ℕ), n ≥ N → ↑↑μ {x | ε ≤ dist (f n x) (g x)} ≤ δ",
"tactic": "simp only [hδi, imp_true_iff, le_top, exists_const]"
}
] |
[
128,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean
|
MulEquiv.ext_iff
|
[] |
[
487,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.nat_lt_lift_iff
|
[
{
"state_after": "no goals",
"state_before": "α β : Type u\nn : ℕ\na : Cardinal\n⊢ ↑n < lift a ↔ ↑n < a",
"tactic": "rw [← lift_natCast.{v,u}, lift_lt]"
}
] |
[
1313,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1312,
1
] |
Mathlib/Data/Polynomial/Module.lean
|
PolynomialModule.eval_map'
|
[] |
[
318,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.prod_quotient_preimage_eq_image
|
[] |
[
576,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean
|
Equiv.Perm.sign_swap
|
[] |
[
579,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
578,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.isMultiplicative_sigma
|
[
{
"state_after": "R : Type ?u.546111\nk : ℕ\n⊢ IsMultiplicative (ζ * pow k)",
"state_before": "R : Type ?u.546111\nk : ℕ\n⊢ IsMultiplicative (σ k)",
"tactic": "rw [← zeta_mul_pow_eq_sigma]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.546111\nk : ℕ\n⊢ IsMultiplicative (ζ * pow k)",
"tactic": "apply isMultiplicative_zeta.mul isMultiplicative_pow"
}
] |
[
841,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
839,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_eq_prod_of_mulSupport_toFinset_subset
|
[] |
[
399,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
UniformContinuous.inf_dom_left
|
[] |
[
1386,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1384,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.AECover.superset
|
[] |
[
119,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
upperBounds_Icc
|
[] |
[
696,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
695,
1
] |
Mathlib/Order/Disjoint.lean
|
eq_bot_of_isCompl_top
|
[] |
[
638,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
637,
1
] |
Mathlib/Topology/FiberBundle/Constructions.lean
|
Trivialization.Prod.left_inv
|
[
{
"state_after": "case mk.mk\nB : Type u_1\ninst✝⁶ : TopologicalSpace B\nF₁ : Type u_4\ninst✝⁵ : TopologicalSpace F₁\nE₁ : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_5\ninst✝³ : TopologicalSpace F₂\nE₂ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : (x : B) → Zero (E₁ x)\ninst✝ : (x : B) → Zero (E₂ x)\nx : B\nv₁ : E₁ x\nv₂ : E₂ x\nh : { fst := x, snd := (v₁, v₂) } ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)\n⊢ invFun' e₁ e₂ (toFun' e₁ e₂ { fst := x, snd := (v₁, v₂) }) = { fst := x, snd := (v₁, v₂) }",
"state_before": "B : Type u_1\ninst✝⁶ : TopologicalSpace B\nF₁ : Type u_4\ninst✝⁵ : TopologicalSpace F₁\nE₁ : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_5\ninst✝³ : TopologicalSpace F₂\nE₂ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : (x : B) → Zero (E₁ x)\ninst✝ : (x : B) → Zero (E₂ x)\nx : TotalSpace fun x => E₁ x × E₂ x\nh : x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)\n⊢ invFun' e₁ e₂ (toFun' e₁ e₂ x) = x",
"tactic": "obtain ⟨x, v₁, v₂⟩ := x"
},
{
"state_after": "case mk.mk.intro\nB : Type u_1\ninst✝⁶ : TopologicalSpace B\nF₁ : Type u_4\ninst✝⁵ : TopologicalSpace F₁\nE₁ : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_5\ninst✝³ : TopologicalSpace F₂\nE₂ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : (x : B) → Zero (E₁ x)\ninst✝ : (x : B) → Zero (E₂ x)\nx : B\nv₁ : E₁ x\nv₂ : E₂ x\nh₁ : x ∈ e₁.baseSet\nh₂ : x ∈ e₂.baseSet\n⊢ invFun' e₁ e₂ (toFun' e₁ e₂ { fst := x, snd := (v₁, v₂) }) = { fst := x, snd := (v₁, v₂) }",
"state_before": "case mk.mk\nB : Type u_1\ninst✝⁶ : TopologicalSpace B\nF₁ : Type u_4\ninst✝⁵ : TopologicalSpace F₁\nE₁ : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_5\ninst✝³ : TopologicalSpace F₂\nE₂ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : (x : B) → Zero (E₁ x)\ninst✝ : (x : B) → Zero (E₂ x)\nx : B\nv₁ : E₁ x\nv₂ : E₂ x\nh : { fst := x, snd := (v₁, v₂) } ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)\n⊢ invFun' e₁ e₂ (toFun' e₁ e₂ { fst := x, snd := (v₁, v₂) }) = { fst := x, snd := (v₁, v₂) }",
"tactic": "obtain ⟨h₁ : x ∈ e₁.baseSet, h₂ : x ∈ e₂.baseSet⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro\nB : Type u_1\ninst✝⁶ : TopologicalSpace B\nF₁ : Type u_4\ninst✝⁵ : TopologicalSpace F₁\nE₁ : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_5\ninst✝³ : TopologicalSpace F₂\nE₂ : B → Type u_2\ninst✝² : TopologicalSpace (TotalSpace E₂)\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : (x : B) → Zero (E₁ x)\ninst✝ : (x : B) → Zero (E₂ x)\nx : B\nv₁ : E₁ x\nv₂ : E₂ x\nh₁ : x ∈ e₁.baseSet\nh₂ : x ∈ e₂.baseSet\n⊢ invFun' e₁ e₂ (toFun' e₁ e₂ { fst := x, snd := (v₁, v₂) }) = { fst := x, snd := (v₁, v₂) }",
"tactic": "simp only [Prod.toFun', Prod.invFun', symm_apply_apply_mk, h₁, h₂]"
}
] |
[
205,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.exists_factorization_lt_of_lt
|
[
{
"state_after": "a b : ℕ\nha : a ≠ 0\nhab : a < b\nhb : b ≠ 0\n⊢ ∃ p, ↑(factorization a) p < ↑(factorization b) p",
"state_before": "a b : ℕ\nha : a ≠ 0\nhab : a < b\n⊢ ∃ p, ↑(factorization a) p < ↑(factorization b) p",
"tactic": "have hb : b ≠ 0 := (ha.bot_lt.trans hab).ne'"
},
{
"state_after": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\nhab : ¬∃ p, ↑(factorization a) p < ↑(factorization b) p\n⊢ b ≤ a",
"state_before": "a b : ℕ\nha : a ≠ 0\nhab : a < b\nhb : b ≠ 0\n⊢ ∃ p, ↑(factorization a) p < ↑(factorization b) p",
"tactic": "contrapose! hab"
},
{
"state_after": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (x : ℕ), ↑(factorization b) x ≤ ↑(factorization a) x\n⊢ b ≤ a",
"state_before": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\nhab : ¬∃ p, ↑(factorization a) p < ↑(factorization b) p\n⊢ b ≤ a",
"tactic": "simp_rw [not_exists, not_lt] at hab"
},
{
"state_after": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\nhab : b ∣ a\n⊢ b ≤ a",
"state_before": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\nhab : ∀ (x : ℕ), ↑(factorization b) x ≤ ↑(factorization a) x\n⊢ b ≤ a",
"tactic": "rw [← Finsupp.le_def, factorization_le_iff_dvd hb ha] at hab"
},
{
"state_after": "no goals",
"state_before": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\nhab : b ∣ a\n⊢ b ≤ a",
"tactic": "exact le_of_dvd ha.bot_lt hab"
}
] |
[
512,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
505,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
IsUnit.stronglyMeasurable_const_smul_iff
|
[] |
[
501,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
8
] |
Mathlib/Topology/ContinuousFunction/Compact.lean
|
ContinuousMap.norm_restrict_mono_set
|
[] |
[
264,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Std/Data/List/Basic.lean
|
List.splitOnP_eq_splitOnPTR
|
[
{
"state_after": "case h.h.h\nα : Type u_1\nP : α → Bool\nl : List α\n⊢ splitOnP P l = splitOnPTR P l",
"state_before": "⊢ @splitOnP = @splitOnPTR",
"tactic": "funext α P l"
},
{
"state_after": "case h.h.h\nα : Type u_1\nP : α → Bool\nl : List α\n⊢ splitOnP P l = splitOnPTR.go P l #[] #[]",
"state_before": "case h.h.h\nα : Type u_1\nP : α → Bool\nl : List α\n⊢ splitOnP P l = splitOnPTR P l",
"tactic": "simp [splitOnPTR]"
},
{
"state_after": "case h.h.h\nα : Type u_1\nP : α → Bool\nl : List α\n⊢ ∀ (xs : List α) (acc : Array α) (r : Array (List α)),\n splitOnPTR.go P xs acc r = r.data ++ splitOnP.go P xs (reverse acc.data)",
"state_before": "case h.h.h\nα : Type u_1\nP : α → Bool\nl : List α\n⊢ splitOnP P l = splitOnPTR.go P l #[] #[]",
"tactic": "suffices ∀ xs acc r, splitOnPTR.go P xs acc r = r.data ++ splitOnP.go P xs acc.data.reverse from\n (this l #[] #[]).symm"
},
{
"state_after": "case h.h.h\nα : Type u_1\nP : α → Bool\nl xs : List α\nacc : Array α\nr : Array (List α)\n⊢ splitOnPTR.go P xs acc r = r.data ++ splitOnP.go P xs (reverse acc.data)",
"state_before": "case h.h.h\nα : Type u_1\nP : α → Bool\nl : List α\n⊢ ∀ (xs : List α) (acc : Array α) (r : Array (List α)),\n splitOnPTR.go P xs acc r = r.data ++ splitOnP.go P xs (reverse acc.data)",
"tactic": "intro xs acc r"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nα : Type u_1\nP : α → Bool\nl xs : List α\nacc : Array α\nr : Array (List α)\n⊢ splitOnPTR.go P xs acc r = r.data ++ splitOnP.go P xs (reverse acc.data)",
"tactic": "induction xs generalizing acc r with simp [splitOnP.go, splitOnPTR.go]\n| cons x xs IH => cases P x <;> simp [*]"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.cons\nα : Type u_1\nP : α → Bool\nl : List α\nx : α\nxs : List α\nIH : ∀ (acc : Array α) (r : Array (List α)), splitOnPTR.go P xs acc r = r.data ++ splitOnP.go P xs (reverse acc.data)\nacc : Array α\nr : Array (List α)\n⊢ (bif P x then splitOnPTR.go P xs #[] (Array.push r acc.data) else splitOnPTR.go P xs (Array.push acc x) r) =\n r.data ++ if P x = true then acc.data :: splitOnP.go P xs [] else splitOnP.go P xs (x :: reverse acc.data)",
"tactic": "cases P x <;> simp [*]"
}
] |
[
499,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
494,
10
] |
Mathlib/Topology/Algebra/Ring/Basic.lean
|
TopologicalRing.of_addGroup_of_nhds_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.17126\nR : Type u_1\ninst✝² : NonUnitalNonAssocRing R\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalAddGroup R\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)\nhmul_left : ∀ (x₀ : R), Tendsto (fun x => x₀ * x) (𝓝 0) (𝓝 0)\nhmul_right : ∀ (x₀ : R), Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)\n⊢ Continuous fun p => p.fst * p.snd",
"tactic": "refine continuous_of_continuousAt_zero₂ (AddMonoidHom.mul (R := R)) ?_ ?_ ?_ <;>\n simpa only [ContinuousAt, mul_zero, zero_mul, nhds_prod_eq, AddMonoidHom.mul_apply]"
}
] |
[
208,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
TopologicalGroup.ext
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG✝ : Type w\nH : Type x\ninst✝⁴ : TopologicalSpace G✝\ninst✝³ : Group G✝\ninst✝² : TopologicalGroup G✝\ninst✝¹ : TopologicalSpace α\nf : α → G✝\ns : Set α\nx✝ : α\nG : Type u_1\ninst✝ : Group G\nt t' : TopologicalSpace G\ntg : TopologicalGroup G\ntg' : TopologicalGroup G\nh : 𝓝 1 = 𝓝 1\nx : G\n⊢ 𝓝 x = 𝓝 x",
"tactic": "rw [← @nhds_translation_mul_inv G t _ _ x, ← @nhds_translation_mul_inv G t' _ _ x, ← h]"
}
] |
[
894,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
890,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.Ioo_cons_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na b : α\nh : a < b\n⊢ a ::ₘ Ioo a b = Ico a b",
"tactic": "classical\n rw [Ioo, ← Finset.insert_val_of_not_mem left_not_mem_Ioo, Finset.Ioo_insert_left h]\n rfl"
},
{
"state_after": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na b : α\nh : a < b\n⊢ (Finset.Ico a b).val = Ico a b",
"state_before": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na b : α\nh : a < b\n⊢ a ::ₘ Ioo a b = Ico a b",
"tactic": "rw [Ioo, ← Finset.insert_val_of_not_mem left_not_mem_Ioo, Finset.Ioo_insert_left h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na b : α\nh : a < b\n⊢ (Finset.Ico a b).val = Ico a b",
"tactic": "rfl"
}
] |
[
206,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.multiplicity_eq_zero
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ a ^ 0 ∣ b ∧ ¬a ^ (0 + 1) ∣ b ↔ ¬a ∣ b",
"state_before": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ multiplicity a b = 0 ↔ ¬a ∣ b",
"tactic": "rw [← Nat.cast_zero, eq_coe_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ a ^ 0 ∣ b ∧ ¬a ^ (0 + 1) ∣ b ↔ ¬a ∣ b",
"tactic": "simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and]"
}
] |
[
206,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/SetTheory/Ordinal/NaturalOps.lean
|
Ordinal.nadd_lt_nadd_iff_right
|
[] |
[
423,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/Topology/Instances/TrivSqZeroExt.lean
|
TrivSqZeroExt.hasSum_snd
|
[] |
[
166,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Analysis/Asymptotics/Theta.lean
|
Asymptotics.IsTheta.trans_isBigO
|
[] |
[
102,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Topology/UniformSpace/CompactConvergence.lean
|
ContinuousMap.hasBasis_compactConvergenceUniformity_aux
|
[
{
"state_after": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\n⊢ DirectedOn ((fun KV => {fg | ∀ (x : α), x ∈ KV.fst → (↑fg.fst x, ↑fg.snd x) ∈ KV.snd}) ⁻¹'o fun x x_1 => x ≥ x_1)\n {KV | IsCompact KV.fst ∧ KV.snd ∈ 𝓤 β}",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\n⊢ HasBasis compactConvergenceUniformity (fun p => IsCompact p.fst ∧ p.snd ∈ 𝓤 β) fun p =>\n {fg | ∀ (x : α), x ∈ p.fst → (↑fg.fst x, ↑fg.snd x) ∈ p.snd}",
"tactic": "refine' Filter.hasBasis_biInf_principal _ compactConvNhd_compact_entourage_nonempty"
},
{
"state_after": "case mk.intro.mk.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nK₁ : Set α\nV₁ : Set (β × β)\nhK₁ : IsCompact (K₁, V₁).fst\nhV₁ : (K₁, V₁).snd ∈ 𝓤 β\nK₂ : Set α\nV₂ : Set (β × β)\nhK₂ : IsCompact (K₂, V₂).fst\nhV₂ : (K₂, V₂).snd ∈ 𝓤 β\n⊢ ∃ z,\n z ∈ {KV | IsCompact KV.fst ∧ KV.snd ∈ 𝓤 β} ∧\n ((fun KV => {fg | ∀ (x : α), x ∈ KV.fst → (↑fg.fst x, ↑fg.snd x) ∈ KV.snd}) ⁻¹'o fun x x_1 => x ≥ x_1) (K₁, V₁)\n z ∧\n ((fun KV => {fg | ∀ (x : α), x ∈ KV.fst → (↑fg.fst x, ↑fg.snd x) ∈ KV.snd}) ⁻¹'o fun x x_1 => x ≥ x_1) (K₂, V₂)\n z",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\n⊢ DirectedOn ((fun KV => {fg | ∀ (x : α), x ∈ KV.fst → (↑fg.fst x, ↑fg.snd x) ∈ KV.snd}) ⁻¹'o fun x x_1 => x ≥ x_1)\n {KV | IsCompact KV.fst ∧ KV.snd ∈ 𝓤 β}",
"tactic": "rintro ⟨K₁, V₁⟩ ⟨hK₁, hV₁⟩ ⟨K₂, V₂⟩ ⟨hK₂, hV₂⟩"
},
{
"state_after": "case mk.intro.mk.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nK₁ : Set α\nV₁ : Set (β × β)\nhK₁ : IsCompact (K₁, V₁).fst\nhV₁ : (K₁, V₁).snd ∈ 𝓤 β\nK₂ : Set α\nV₂ : Set (β × β)\nhK₂ : IsCompact (K₂, V₂).fst\nhV₂ : (K₂, V₂).snd ∈ 𝓤 β\n⊢ ((fun KV => {fg | ∀ (x : α), x ∈ KV.fst → (↑fg.fst x, ↑fg.snd x) ∈ KV.snd}) ⁻¹'o fun x x_1 => x ≥ x_1) (K₁, V₁)\n (K₁ ∪ K₂, V₁ ∩ V₂) ∧\n ((fun KV => {fg | ∀ (x : α), x ∈ KV.fst → (↑fg.fst x, ↑fg.snd x) ∈ KV.snd}) ⁻¹'o fun x x_1 => x ≥ x_1) (K₂, V₂)\n (K₁ ∪ K₂, V₁ ∩ V₂)",
"state_before": "case mk.intro.mk.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nK₁ : Set α\nV₁ : Set (β × β)\nhK₁ : IsCompact (K₁, V₁).fst\nhV₁ : (K₁, V₁).snd ∈ 𝓤 β\nK₂ : Set α\nV₂ : Set (β × β)\nhK₂ : IsCompact (K₂, V₂).fst\nhV₂ : (K₂, V₂).snd ∈ 𝓤 β\n⊢ ∃ z,\n z ∈ {KV | IsCompact KV.fst ∧ KV.snd ∈ 𝓤 β} ∧\n ((fun KV => {fg | ∀ (x : α), x ∈ KV.fst → (↑fg.fst x, ↑fg.snd x) ∈ KV.snd}) ⁻¹'o fun x x_1 => x ≥ x_1) (K₁, V₁)\n z ∧\n ((fun KV => {fg | ∀ (x : α), x ∈ KV.fst → (↑fg.fst x, ↑fg.snd x) ∈ KV.snd}) ⁻¹'o fun x x_1 => x ≥ x_1) (K₂, V₂)\n z",
"tactic": "refine' ⟨⟨K₁ ∪ K₂, V₁ ∩ V₂⟩, ⟨hK₁.union hK₂, Filter.inter_mem hV₁ hV₂⟩, _⟩"
},
{
"state_after": "case mk.intro.mk.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nK₁ : Set α\nV₁ : Set (β × β)\nhK₁ : IsCompact (K₁, V₁).fst\nhV₁ : (K₁, V₁).snd ∈ 𝓤 β\nK₂ : Set α\nV₂ : Set (β × β)\nhK₂ : IsCompact (K₂, V₂).fst\nhV₂ : (K₂, V₂).snd ∈ 𝓤 β\n⊢ ∀ (x x_1 : C(α, β)),\n (∀ (x_2 : α), x_2 ∈ K₁ ∨ x_2 ∈ K₂ → (↑x x_2, ↑x_1 x_2) ∈ V₁ ∧ (↑x x_2, ↑x_1 x_2) ∈ V₂) →\n ∀ (x_3 : α), (x_3 ∈ K₁ → (↑x x_3, ↑x_1 x_3) ∈ V₁) ∧ (x_3 ∈ K₂ → (↑x x_3, ↑x_1 x_3) ∈ V₂)",
"state_before": "case mk.intro.mk.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nK₁ : Set α\nV₁ : Set (β × β)\nhK₁ : IsCompact (K₁, V₁).fst\nhV₁ : (K₁, V₁).snd ∈ 𝓤 β\nK₂ : Set α\nV₂ : Set (β × β)\nhK₂ : IsCompact (K₂, V₂).fst\nhV₂ : (K₂, V₂).snd ∈ 𝓤 β\n⊢ ((fun KV => {fg | ∀ (x : α), x ∈ KV.fst → (↑fg.fst x, ↑fg.snd x) ∈ KV.snd}) ⁻¹'o fun x x_1 => x ≥ x_1) (K₁, V₁)\n (K₁ ∪ K₂, V₁ ∩ V₂) ∧\n ((fun KV => {fg | ∀ (x : α), x ∈ KV.fst → (↑fg.fst x, ↑fg.snd x) ∈ KV.snd}) ⁻¹'o fun x x_1 => x ≥ x_1) (K₂, V₂)\n (K₁ ∪ K₂, V₁ ∩ V₂)",
"tactic": "simp only [le_eq_subset, Prod.forall, setOf_subset_setOf, ge_iff_le, Order.Preimage, ←\n forall_and, mem_inter_iff, mem_union]"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.mk.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nK₁ : Set α\nV₁ : Set (β × β)\nhK₁ : IsCompact (K₁, V₁).fst\nhV₁ : (K₁, V₁).snd ∈ 𝓤 β\nK₂ : Set α\nV₂ : Set (β × β)\nhK₂ : IsCompact (K₂, V₂).fst\nhV₂ : (K₂, V₂).snd ∈ 𝓤 β\n⊢ ∀ (x x_1 : C(α, β)),\n (∀ (x_2 : α), x_2 ∈ K₁ ∨ x_2 ∈ K₂ → (↑x x_2, ↑x_1 x_2) ∈ V₁ ∧ (↑x x_2, ↑x_1 x_2) ∈ V₂) →\n ∀ (x_3 : α), (x_3 ∈ K₁ → (↑x x_3, ↑x_1 x_3) ∈ V₁) ∧ (x_3 ∈ K₂ → (↑x x_3, ↑x_1 x_3) ∈ V₂)",
"tactic": "exact fun f g => forall_imp fun x => by tauto"
},
{
"state_after": "no goals",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf✝ : C(α, β)\nK₁ : Set α\nV₁ : Set (β × β)\nhK₁ : IsCompact (K₁, V₁).fst\nhV₁ : (K₁, V₁).snd ∈ 𝓤 β\nK₂ : Set α\nV₂ : Set (β × β)\nhK₂ : IsCompact (K₂, V₂).fst\nhV₂ : (K₂, V₂).snd ∈ 𝓤 β\nf g : C(α, β)\nx : α\n⊢ (x ∈ K₁ ∨ x ∈ K₂ → (↑f x, ↑g x) ∈ V₁ ∧ (↑f x, ↑g x) ∈ V₂) →\n (x ∈ K₁ → (↑f x, ↑g x) ∈ V₁) ∧ (x ∈ K₂ → (↑f x, ↑g x) ∈ V₂)",
"tactic": "tauto"
}
] |
[
307,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/Topology/Basic.lean
|
DenseRange.nonempty
|
[] |
[
1858,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1857,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Ideal.is_fg_degreeLE
|
[] |
[
736,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
732,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.num_div_dvd
|
[
{
"state_after": "K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ p / gcd p q ∣ p\n\nK : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ (leadingCoeff (q / gcd p q))⁻¹ ≠ 0",
"state_before": "K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ num (↑(algebraMap K[X] (RatFunc K)) p / ↑(algebraMap K[X] (RatFunc K)) q) ∣ p",
"tactic": "rw [num_div _ q, C_mul_dvd]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ p / gcd p q ∣ p",
"tactic": "exact EuclideanDomain.div_dvd_of_dvd (gcd_dvd_left p q)"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ (leadingCoeff (q / gcd p q))⁻¹ ≠ 0",
"tactic": "simpa only [Ne.def, inv_eq_zero, Polynomial.leadingCoeff_eq_zero] using right_div_gcd_ne_zero hq"
}
] |
[
1177,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1173,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.isClosed_image_coe_closedBall
|
[] |
[
1662,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1660,
1
] |
Mathlib/CategoryTheory/Preadditive/Basic.lean
|
CategoryTheory.Preadditive.hasCoequalizers_of_hasCokernels
|
[] |
[
444,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.bot_prod_bot
|
[] |
[
1572,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1571,
1
] |
Mathlib/GroupTheory/Subsemigroup/Center.lean
|
Set.center_eq_univ
|
[] |
[
134,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Topology/Order.lean
|
continuous_sInf_dom
|
[] |
[
790,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
787,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.pow_mem
|
[] |
[
137,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
11
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.mem_sSup
|
[] |
[
568,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/Data/PNat/Prime.lean
|
Nat.Primes.coe_pnat_nat
|
[] |
[
35,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Mathlib/Data/Polynomial/EraseLead.lean
|
Polynomial.eraseLead_card_support'
|
[] |
[
126,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/Int/Lemmas.lean
|
Int.le_coe_nat_sub
|
[
{
"state_after": "case pos\nm n : ℕ\nh : m ≥ n\n⊢ ↑m - ↑n ≤ ↑(m - n)\n\ncase neg\nm n : ℕ\nh : ¬m ≥ n\n⊢ ↑m - ↑n ≤ ↑(m - n)",
"state_before": "m n : ℕ\n⊢ ↑m - ↑n ≤ ↑(m - n)",
"tactic": "by_cases h : m ≥ n"
},
{
"state_after": "no goals",
"state_before": "case pos\nm n : ℕ\nh : m ≥ n\n⊢ ↑m - ↑n ≤ ↑(m - n)",
"tactic": "exact le_of_eq (Int.ofNat_sub h).symm"
},
{
"state_after": "no goals",
"state_before": "case neg\nm n : ℕ\nh : ¬m ≥ n\n⊢ ↑m - ↑n ≤ ↑(m - n)",
"tactic": "simp [le_of_not_ge h, ofNat_le]"
}
] |
[
33,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/Data/Int/Order/Basic.lean
|
Int.le_ediv_iff_mul_le
|
[] |
[
443,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
11
] |
Mathlib/LinearAlgebra/Matrix/Trace.lean
|
Matrix.trace_fin_two
|
[] |
[
195,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
HomologicalComplex.Hom.comm
|
[
{
"state_after": "case pos\nι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC A B : HomologicalComplex V c\nf : Hom A B\ni j : ι\nhij : ComplexShape.Rel c i j\n⊢ HomologicalComplex.Hom.f f i ≫ d B i j = d A i j ≫ HomologicalComplex.Hom.f f j\n\ncase neg\nι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC A B : HomologicalComplex V c\nf : Hom A B\ni j : ι\nhij : ¬ComplexShape.Rel c i j\n⊢ HomologicalComplex.Hom.f f i ≫ d B i j = d A i j ≫ HomologicalComplex.Hom.f f j",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC A B : HomologicalComplex V c\nf : Hom A B\ni j : ι\n⊢ HomologicalComplex.Hom.f f i ≫ d B i j = d A i j ≫ HomologicalComplex.Hom.f f j",
"tactic": "by_cases hij : c.Rel i j"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC A B : HomologicalComplex V c\nf : Hom A B\ni j : ι\nhij : ¬ComplexShape.Rel c i j\n⊢ HomologicalComplex.Hom.f f i ≫ d B i j = d A i j ≫ HomologicalComplex.Hom.f f j",
"tactic": ". rw [A.shape i j hij, B.shape i j hij, comp_zero, zero_comp]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC A B : HomologicalComplex V c\nf : Hom A B\ni j : ι\nhij : ComplexShape.Rel c i j\n⊢ HomologicalComplex.Hom.f f i ≫ d B i j = d A i j ≫ HomologicalComplex.Hom.f f j",
"tactic": "exact f.comm' i j hij"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC A B : HomologicalComplex V c\nf : Hom A B\ni j : ι\nhij : ¬ComplexShape.Rel c i j\n⊢ HomologicalComplex.Hom.f f i ≫ d B i j = d A i j ≫ HomologicalComplex.Hom.f f j",
"tactic": "rw [A.shape i j hij, B.shape i j hij, comp_zero, zero_comp]"
}
] |
[
191,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/CategoryTheory/Balanced.lean
|
CategoryTheory.isIso_iff_mono_and_epi
|
[] |
[
47,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Algebra/Hom/GroupAction.lean
|
MulActionHom.comp_id
|
[
{
"state_after": "no goals",
"state_before": "M' : Type u_1\nX : Type u_2\ninst✝²³ : SMul M' X\nY : Type u_3\ninst✝²² : SMul M' Y\nZ : Type ?u.48568\ninst✝²¹ : SMul M' Z\nM : Type ?u.48575\ninst✝²⁰ : Monoid M\nA : Type ?u.48581\ninst✝¹⁹ : AddMonoid A\ninst✝¹⁸ : DistribMulAction M A\nA' : Type ?u.48609\ninst✝¹⁷ : AddGroup A'\ninst✝¹⁶ : DistribMulAction M A'\nB : Type ?u.48885\ninst✝¹⁵ : AddMonoid B\ninst✝¹⁴ : DistribMulAction M B\nB' : Type ?u.48911\ninst✝¹³ : AddGroup B'\ninst✝¹² : DistribMulAction M B'\nC : Type ?u.49187\ninst✝¹¹ : AddMonoid C\ninst✝¹⁰ : DistribMulAction M C\nR : Type ?u.49213\ninst✝⁹ : Semiring R\ninst✝⁸ : MulSemiringAction M R\nR' : Type ?u.49240\ninst✝⁷ : Ring R'\ninst✝⁶ : MulSemiringAction M R'\nS : Type ?u.49436\ninst✝⁵ : Semiring S\ninst✝⁴ : MulSemiringAction M S\nS' : Type ?u.49463\ninst✝³ : Ring S'\ninst✝² : MulSemiringAction M S'\nT : Type ?u.49659\ninst✝¹ : Semiring T\ninst✝ : MulSemiringAction M T\nf : X →[M'] Y\nx : X\n⊢ ↑(comp f (MulActionHom.id M')) x = ↑f x",
"tactic": "rw [comp_apply, id_apply]"
}
] |
[
171,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
affineCombination_mem_affineSpan_pair
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nx✝ : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\n⊢ (∃ r, ∀ (i : ι), i ∈ s → (w - w₁) i = r * (w₂ i - w₁ i)) ↔ ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i\n\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nx✝ : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\n⊢ ∑ i in s, (w - w₁) i = 0",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nx✝ : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\n⊢ ↑(Finset.affineCombination k s p) w ∈\n affineSpan k {↑(Finset.affineCombination k s p) w₁, ↑(Finset.affineCombination k s p) w₂} ↔\n ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i",
"tactic": "rw [← vsub_vadd (s.affineCombination k p w) (s.affineCombination k p w₁),\n AffineSubspace.vadd_mem_iff_mem_direction _ (left_mem_affineSpan_pair _ _ _),\n direction_affineSpan, s.affineCombination_vsub, Set.pair_comm,\n weightedVSub_mem_vectorSpan_pair h _ hw₂ hw₁]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nx✝ : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\n⊢ (∃ r, ∀ (i : ι), i ∈ s → (w - w₁) i = r * (w₂ i - w₁ i)) ↔ ∃ r, ∀ (i : ι), i ∈ s → w i = r * (w₂ i - w₁ i) + w₁ i",
"tactic": "simp only [Pi.sub_apply, sub_eq_iff_eq_add]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw w₁ w₂ : ι → k\ns : Finset ι\nx✝ : ∑ i in s, w i = 1\nhw₁ : ∑ i in s, w₁ i = 1\nhw₂ : ∑ i in s, w₂ i = 1\n⊢ ∑ i in s, (w - w₁) i = 0",
"tactic": "simp_all only [Pi.sub_apply, Finset.sum_sub_distrib, sub_self]"
}
] |
[
539,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrixAlgEquiv_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁹ : CommSemiring R\nl : Type ?u.2330336\nm : Type ?u.2330339\nn : Type u_3\ninst✝⁸ : Fintype n\ninst✝⁷ : Fintype m\ninst✝⁶ : DecidableEq n\nM₁ : Type u_2\nM₂ : Type ?u.2330363\ninst✝⁵ : AddCommMonoid M₁\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R M₁\ninst✝² : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nM₃ : Type ?u.2330865\ninst✝¹ : AddCommMonoid M₃\ninst✝ : Module R M₃\nv₃ : Basis l R M₃\nf : M₁ →ₗ[R] M₁\ni j : n\n⊢ ↑(toMatrixAlgEquiv v₁) f i j = ↑(↑v₁.repr (↑f (↑v₁ j))) i",
"tactic": "simp [LinearMap.toMatrixAlgEquiv, LinearMap.toMatrix_apply]"
}
] |
[
748,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
746,
1
] |
Mathlib/Computability/Primrec.lean
|
Nat.Primrec'.cons
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\nf : Vector ℕ n → ℕ\ng : Vector ℕ n → Vector ℕ m\nhf : Primrec' f\nhg : Vec g\ni : Fin (Nat.succ m)\n⊢ Primrec' fun v => Vector.get ((fun v => f v ::ᵥ g v) v) 0",
"tactic": "simp [*]"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ\nf : Vector ℕ n → ℕ\ng : Vector ℕ n → Vector ℕ m\nhf : Primrec' f\nhg : Vec g\ni✝ : Fin (Nat.succ m)\ni : Fin m\n⊢ Primrec' fun v => Vector.get ((fun v => f v ::ᵥ g v) v) (Fin.succ i)",
"tactic": "simp [hg i]"
}
] |
[
1424,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1423,
11
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.IsImage.leftInvOn_piecewise
|
[] |
[
570,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.cthickening_eq_iInter_cthickening'
|
[
{
"state_after": "case h₁\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\n⊢ cthickening δ E ⊆ ⋂ (ε : ℝ) (_ : ε ∈ s), cthickening ε E\n\ncase h₂\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\n⊢ (⋂ (ε : ℝ) (_ : ε ∈ s), cthickening ε E) ⊆ cthickening δ E",
"state_before": "ι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\n⊢ cthickening δ E = ⋂ (ε : ℝ) (_ : ε ∈ s), cthickening ε E",
"tactic": "apply Subset.antisymm"
},
{
"state_after": "no goals",
"state_before": "case h₁\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\n⊢ cthickening δ E ⊆ ⋂ (ε : ℝ) (_ : ε ∈ s), cthickening ε E",
"tactic": "exact subset_iInter₂ fun _ hε => cthickening_mono (le_of_lt (hsδ hε)) E"
},
{
"state_after": "case h₂\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\n⊢ (⋂ (ε : ℝ) (_ : ε ∈ s), {x | infEdist x E ≤ ENNReal.ofReal ε}) ⊆ {x | infEdist x E ≤ ENNReal.ofReal δ}",
"state_before": "case h₂\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\n⊢ (⋂ (ε : ℝ) (_ : ε ∈ s), cthickening ε E) ⊆ cthickening δ E",
"tactic": "unfold cthickening"
},
{
"state_after": "case h₂\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : x ∈ ⋂ (ε : ℝ) (_ : ε ∈ s), {x | infEdist x E ≤ ENNReal.ofReal ε}\n⊢ x ∈ {x | infEdist x E ≤ ENNReal.ofReal δ}",
"state_before": "case h₂\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\n⊢ (⋂ (ε : ℝ) (_ : ε ∈ s), {x | infEdist x E ≤ ENNReal.ofReal ε}) ⊆ {x | infEdist x E ≤ ENNReal.ofReal δ}",
"tactic": "intro x hx"
},
{
"state_after": "case h₂\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\n⊢ infEdist x E ≤ ENNReal.ofReal δ",
"state_before": "case h₂\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : x ∈ ⋂ (ε : ℝ) (_ : ε ∈ s), {x | infEdist x E ≤ ENNReal.ofReal ε}\n⊢ x ∈ {x | infEdist x E ≤ ENNReal.ofReal δ}",
"tactic": "simp only [mem_iInter, mem_setOf_eq] at *"
},
{
"state_after": "case h₂.h\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\n⊢ ∀ (ε : ℝ≥0), 0 < ε → ENNReal.ofReal δ < ⊤ → infEdist x E ≤ ENNReal.ofReal δ + ↑ε",
"state_before": "case h₂\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\n⊢ infEdist x E ≤ ENNReal.ofReal δ",
"tactic": "apply ENNReal.le_of_forall_pos_le_add"
},
{
"state_after": "case h₂.h\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\nη : ℝ≥0\nη_pos : 0 < η\na✝ : ENNReal.ofReal δ < ⊤\n⊢ infEdist x E ≤ ENNReal.ofReal δ + ↑η",
"state_before": "case h₂.h\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\n⊢ ∀ (ε : ℝ≥0), 0 < ε → ENNReal.ofReal δ < ⊤ → infEdist x E ≤ ENNReal.ofReal δ + ↑ε",
"tactic": "intro η η_pos _"
},
{
"state_after": "case h₂.h.intro.intro\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\nη : ℝ≥0\nη_pos : 0 < η\na✝ : ENNReal.ofReal δ < ⊤\nε : ℝ\nhsε : ε ∈ s\nhε : ε ∈ Ioc δ (δ + ↑η)\n⊢ infEdist x E ≤ ENNReal.ofReal δ + ↑η",
"state_before": "case h₂.h\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\nη : ℝ≥0\nη_pos : 0 < η\na✝ : ENNReal.ofReal δ < ⊤\n⊢ infEdist x E ≤ ENNReal.ofReal δ + ↑η",
"tactic": "rcases hs (δ + η) (lt_add_of_pos_right _ (NNReal.coe_pos.mpr η_pos)) with ⟨ε, ⟨hsε, hε⟩⟩"
},
{
"state_after": "case h₂.h.intro.intro\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\nη : ℝ≥0\nη_pos : 0 < η\na✝ : ENNReal.ofReal δ < ⊤\nε : ℝ\nhsε : ε ∈ s\nhε : ε ∈ Ioc δ (δ + ↑η)\n⊢ ENNReal.ofReal (δ + ↑η) ≤ ENNReal.ofReal δ + ↑η",
"state_before": "case h₂.h.intro.intro\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\nη : ℝ≥0\nη_pos : 0 < η\na✝ : ENNReal.ofReal δ < ⊤\nε : ℝ\nhsε : ε ∈ s\nhε : ε ∈ Ioc δ (δ + ↑η)\n⊢ infEdist x E ≤ ENNReal.ofReal δ + ↑η",
"tactic": "apply ((hx ε hsε).trans (ENNReal.ofReal_le_ofReal hε.2)).trans"
},
{
"state_after": "case h₂.h.intro.intro\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\nη : ℝ≥0\nη_pos : 0 < η\na✝ : ENNReal.ofReal δ < ⊤\nε : ℝ\nhsε : ε ∈ s\nhε : ε ∈ Ioc δ (δ + ↑η)\n⊢ ENNReal.ofReal (δ + ↑η) ≤ ENNReal.ofReal δ + ENNReal.ofReal ↑η",
"state_before": "case h₂.h.intro.intro\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\nη : ℝ≥0\nη_pos : 0 < η\na✝ : ENNReal.ofReal δ < ⊤\nε : ℝ\nhsε : ε ∈ s\nhε : ε ∈ Ioc δ (δ + ↑η)\n⊢ ENNReal.ofReal (δ + ↑η) ≤ ENNReal.ofReal δ + ↑η",
"tactic": "rw [ENNReal.coe_nnreal_eq η]"
},
{
"state_after": "no goals",
"state_before": "case h₂.h.intro.intro\nι : Sort ?u.139615\nα : Type u\nβ : Type v\ninst✝ : PseudoEMetricSpace α\nδ✝ ε✝ : ℝ\ns✝ t : Set α\nx✝ : α\nδ : ℝ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → Set.Nonempty (s ∩ Ioc δ ε)\nE : Set α\nx : α\nhx : ∀ (i : ℝ), i ∈ s → infEdist x E ≤ ENNReal.ofReal i\nη : ℝ≥0\nη_pos : 0 < η\na✝ : ENNReal.ofReal δ < ⊤\nε : ℝ\nhsε : ε ∈ s\nhε : ε ∈ Ioc δ (δ + ↑η)\n⊢ ENNReal.ofReal (δ + ↑η) ≤ ENNReal.ofReal δ + ENNReal.ofReal ↑η",
"tactic": "exact ENNReal.ofReal_add_le"
}
] |
[
1310,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1297,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurable_of_finite
|
[] |
[
296,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_le_rpow_left_iff
|
[
{
"state_after": "x y z : ℝ\nhx : 1 < x\nx_pos : 0 < x\n⊢ x ^ y ≤ x ^ z ↔ y ≤ z",
"state_before": "x y z : ℝ\nhx : 1 < x\n⊢ x ^ y ≤ x ^ z ↔ y ≤ z",
"tactic": "have x_pos : 0 < x := lt_trans zero_lt_one hx"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhx : 1 < x\nx_pos : 0 < x\n⊢ x ^ y ≤ x ^ z ↔ y ≤ z",
"tactic": "rw [← log_le_log (rpow_pos_of_pos x_pos y) (rpow_pos_of_pos x_pos z), log_rpow x_pos,\n log_rpow x_pos, mul_le_mul_right (log_pos hx)]"
}
] |
[
484,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/RingTheory/RootsOfUnity/Complex.lean
|
IsPrimitiveRoot.arg_eq_pi_iff
|
[] |
[
132,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.mulIndicator_iUnion_apply
|
[
{
"state_after": "case pos\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : x ∈ ⋃ (i : ι), s i\n⊢ mulIndicator (⋃ (i : ι), s i) f x = ⨆ (i : ι), mulIndicator (s i) f x\n\ncase neg\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : ¬x ∈ ⋃ (i : ι), s i\n⊢ mulIndicator (⋃ (i : ι), s i) f x = ⨆ (i : ι), mulIndicator (s i) f x",
"state_before": "α : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\n⊢ mulIndicator (⋃ (i : ι), s i) f x = ⨆ (i : ι), mulIndicator (s i) f x",
"tactic": "by_cases hx : x ∈ ⋃ i, s i"
},
{
"state_after": "case pos\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : x ∈ ⋃ (i : ι), s i\n⊢ f x = ⨆ (i : ι), mulIndicator (s i) f x",
"state_before": "case pos\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : x ∈ ⋃ (i : ι), s i\n⊢ mulIndicator (⋃ (i : ι), s i) f x = ⨆ (i : ι), mulIndicator (s i) f x",
"tactic": "rw [mulIndicator_of_mem hx]"
},
{
"state_after": "case pos\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : ∃ i, x ∈ s i\n⊢ f x = ⨆ (i : ι), mulIndicator (s i) f x",
"state_before": "case pos\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : x ∈ ⋃ (i : ι), s i\n⊢ f x = ⨆ (i : ι), mulIndicator (s i) f x",
"tactic": "rw [mem_iUnion] at hx"
},
{
"state_after": "case pos\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : ∃ i, x ∈ s i\n⊢ f x ≤ ⨆ (i : ι), mulIndicator (s i) f x",
"state_before": "case pos\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : ∃ i, x ∈ s i\n⊢ f x = ⨆ (i : ι), mulIndicator (s i) f x",
"tactic": "refine' le_antisymm _ (iSup_le fun i => mulIndicator_le_self' (fun x _ => h1 ▸ bot_le) x)"
},
{
"state_after": "case pos.intro\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\ni : ι\nhi : x ∈ s i\n⊢ f x ≤ ⨆ (i : ι), mulIndicator (s i) f x",
"state_before": "case pos\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : ∃ i, x ∈ s i\n⊢ f x ≤ ⨆ (i : ι), mulIndicator (s i) f x",
"tactic": "rcases hx with ⟨i, hi⟩"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\ni : ι\nhi : x ∈ s i\n⊢ f x ≤ ⨆ (i : ι), mulIndicator (s i) f x",
"tactic": "exact le_iSup_of_le i (ge_of_eq <| mulIndicator_of_mem hi _)"
},
{
"state_after": "case neg\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : ¬x ∈ ⋃ (i : ι), s i\n⊢ 1 = ⨆ (i : ι), mulIndicator (s i) f x",
"state_before": "case neg\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : ¬x ∈ ⋃ (i : ι), s i\n⊢ mulIndicator (⋃ (i : ι), s i) f x = ⨆ (i : ι), mulIndicator (s i) f x",
"tactic": "rw [mulIndicator_of_not_mem hx]"
},
{
"state_after": "case neg\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : ∀ (x_1 : ι), ¬x ∈ s x_1\n⊢ 1 = ⨆ (i : ι), mulIndicator (s i) f x",
"state_before": "case neg\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : ¬x ∈ ⋃ (i : ι), s i\n⊢ 1 = ⨆ (i : ι), mulIndicator (s i) f x",
"tactic": "simp only [mem_iUnion, not_exists] at hx"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_3\nβ : Type ?u.157531\nι✝ : Type ?u.157534\nM✝ : Type ?u.157537\nN : Type ?u.157540\ninst✝³ : One M✝\ns✝ t : Set α\nf✝ g : α → M✝\na : α\ny : M✝\ninst✝² : Preorder M✝\nι : Sort u_1\nM : Type u_2\ninst✝¹ : CompleteLattice M\ninst✝ : One M\nh1 : ⊥ = 1\ns : ι → Set α\nf : α → M\nx : α\nhx : ∀ (x_1 : ι), ¬x ∈ s x_1\n⊢ 1 = ⨆ (i : ι), mulIndicator (s i) f x",
"tactic": "simp [hx, ← h1]"
}
] |
[
849,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
838,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.aleph'_cof
|
[] |
[
726,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
725,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
NNReal.rpow_inv_rpow_self
|
[
{
"state_after": "no goals",
"state_before": "y : ℝ\nhy : y ≠ 0\nx : ℝ≥0\n⊢ (x ^ y) ^ (1 / y) = x",
"tactic": "field_simp [← rpow_mul]"
}
] |
[
108,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.exists_closedBall_inter_eq_singleton_of_discrete
|
[] |
[
1129,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1127,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.rootSet_prod
|
[
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝³ : Field R\np q : R[X]\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nι : Type u_1\nf : ι → R[X]\ns : Finset ι\nh : Finset.prod s f ≠ 0\n⊢ ↑(Multiset.toFinset (roots (map (algebraMap R S) (Finset.prod s f)))) =\n ⋃ (i : ι) (_ : i ∈ ↑s), ↑(Multiset.toFinset (roots (map (algebraMap R S) (f i))))",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝³ : Field R\np q : R[X]\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nι : Type u_1\nf : ι → R[X]\ns : Finset ι\nh : Finset.prod s f ≠ 0\n⊢ rootSet (Finset.prod s f) S = ⋃ (i : ι) (_ : i ∈ s), rootSet (f i) S",
"tactic": "simp only [rootSet, ← Finset.mem_coe]"
},
{
"state_after": "case a\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝³ : Field R\np q : R[X]\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nι : Type u_1\nf : ι → R[X]\ns : Finset ι\nh : Finset.prod s f ≠ 0\n⊢ ∏ i in s, map (algebraMap R S) (f i) ≠ 0",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝³ : Field R\np q : R[X]\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nι : Type u_1\nf : ι → R[X]\ns : Finset ι\nh : Finset.prod s f ≠ 0\n⊢ ↑(Multiset.toFinset (roots (map (algebraMap R S) (Finset.prod s f)))) =\n ⋃ (i : ι) (_ : i ∈ ↑s), ↑(Multiset.toFinset (roots (map (algebraMap R S) (f i))))",
"tactic": "rw [Polynomial.map_prod, roots_prod, Finset.bind_toFinset, s.val_toFinset, Finset.coe_biUnion]"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝³ : Field R\np q : R[X]\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nι : Type u_1\nf : ι → R[X]\ns : Finset ι\nh : Finset.prod s f ≠ 0\n⊢ ∏ i in s, map (algebraMap R S) (f i) ≠ 0",
"tactic": "rwa [← Polynomial.map_prod, Ne, map_eq_zero]"
}
] |
[
391,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.attach_cons
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.143075\nγ : Type ?u.143078\na : α\nm : Multiset α\nl : List α\n⊢ List.pmap mk l (_ : ∀ (x : α), x ∈ l → (fun a_2 => a_2 ∈ a ::ₘ Quotient.mk (isSetoid α) l) x) =\n List.pmap\n (fun a_1 h =>\n { val := ↑{ val := a_1, property := h },\n property := (_ : ↑{ val := a_1, property := h } ∈ a ::ₘ Quotient.mk (isSetoid α) l) })\n l (_ : ∀ (_a : α), _a ∈ Quotient.mk (isSetoid α) l → _a ∈ Quotient.mk (isSetoid α) l)",
"state_before": "α : Type u_1\nβ : Type ?u.143075\nγ : Type ?u.143078\na : α\nm : Multiset α\nl : List α\n⊢ List.pmap mk l (_ : ∀ (x : α), x ∈ l → (fun a_2 => a_2 ∈ a ::ₘ Quotient.mk (isSetoid α) l) x) =\n List.map (fun p => { val := ↑p, property := (_ : ↑p ∈ a ::ₘ Quotient.mk (isSetoid α) l) })\n (List.pmap mk l (_ : ∀ (_a : α), _a ∈ Quotient.mk (isSetoid α) l → _a ∈ Quotient.mk (isSetoid α) l))",
"tactic": "rw [List.map_pmap]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.143075\nγ : Type ?u.143078\na : α\nm : Multiset α\nl : List α\n⊢ List.pmap mk l (_ : ∀ (x : α), x ∈ l → (fun a_2 => a_2 ∈ a ::ₘ Quotient.mk (isSetoid α) l) x) =\n List.pmap\n (fun a_1 h =>\n { val := ↑{ val := a_1, property := h },\n property := (_ : ↑{ val := a_1, property := h } ∈ a ::ₘ Quotient.mk (isSetoid α) l) })\n l (_ : ∀ (_a : α), _a ∈ Quotient.mk (isSetoid α) l → _a ∈ Quotient.mk (isSetoid α) l)",
"tactic": "exact List.pmap_congr _ fun _ _ _ _ => Subtype.eq rfl"
}
] |
[
1577,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1571,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.ofFinsupp_pow
|
[
{
"state_after": "R : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\na : AddMonoidAlgebra R ℕ\nn : ℕ\n⊢ { toFinsupp := a ^ n } = npowRec n { toFinsupp := a }",
"state_before": "R : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\na : AddMonoidAlgebra R ℕ\nn : ℕ\n⊢ { toFinsupp := a ^ n } = { toFinsupp := a } ^ n",
"tactic": "change _ = npowRec n _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\na : AddMonoidAlgebra R ℕ\nn : ℕ\n⊢ { toFinsupp := a ^ n } = npowRec n { toFinsupp := a }",
"tactic": "induction n with\n| zero => simp [npowRec]\n| succ n n_ih => simp [npowRec, n_ih, pow_succ]"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\na✝ b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\na : AddMonoidAlgebra R ℕ\n⊢ { toFinsupp := a ^ Nat.zero } = npowRec Nat.zero { toFinsupp := a }",
"tactic": "simp [npowRec]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\na : AddMonoidAlgebra R ℕ\nn : ℕ\nn_ih : { toFinsupp := a ^ n } = npowRec n { toFinsupp := a }\n⊢ { toFinsupp := a ^ Nat.succ n } = npowRec (Nat.succ n) { toFinsupp := a }",
"tactic": "simp [npowRec, n_ih, pow_succ]"
}
] |
[
198,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.left_inv'
|
[] |
[
171,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Order/Antichain.lean
|
isAntichain_insert_of_symmetric
|
[] |
[
115,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
convex_singleton
|
[] |
[
175,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Algebra/Support.lean
|
Function.mulSupport_comp_eq_preimage
|
[] |
[
223,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.mem_cycleType_iff
|
[
{
"state_after": "case mp\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nσ : Perm α\n⊢ n ∈ cycleType σ → ∃ c τ, σ = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n\n\ncase mpr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nσ : Perm α\n⊢ (∃ c τ, σ = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n) → n ∈ cycleType σ",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nσ : Perm α\n⊢ n ∈ cycleType σ ↔ ∃ c τ, σ = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nσ : Perm α\nh : n ∈ cycleType σ\n⊢ ∃ c τ, σ = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n",
"state_before": "case mp\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nσ : Perm α\n⊢ n ∈ cycleType σ → ∃ c τ, σ = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n",
"tactic": "intro h"
},
{
"state_after": "case mp.mk.mk.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nl : List (Perm α)\nh : n ∈ cycleType (List.prod l)\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nhld : List.Pairwise Disjoint l\n⊢ ∃ c τ, List.prod l = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n",
"state_before": "case mp\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nσ : Perm α\nh : n ∈ cycleType σ\n⊢ ∃ c τ, σ = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n",
"tactic": "obtain ⟨l, rfl, hlc, hld⟩ := truncCycleFactors σ"
},
{
"state_after": "case mp.mk.mk.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nl : List (Perm α)\nh : ∃ a, a ∈ l ∧ (Finset.card ∘ support) a = n\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nhld : List.Pairwise Disjoint l\n⊢ ∃ c τ, List.prod l = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n",
"state_before": "case mp.mk.mk.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nl : List (Perm α)\nh : n ∈ cycleType (List.prod l)\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nhld : List.Pairwise Disjoint l\n⊢ ∃ c τ, List.prod l = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n",
"tactic": "rw [cycleType_eq _ rfl hlc hld, Multiset.mem_coe, List.mem_map] at h"
},
{
"state_after": "case mp.mk.mk.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nl : List (Perm α)\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nhld : List.Pairwise Disjoint l\nc : Perm α\ncl : c ∈ l\n⊢ ∃ c_1 τ, List.prod l = c_1 * τ ∧ Disjoint c_1 τ ∧ IsCycle c_1 ∧ Finset.card (support c_1) = (Finset.card ∘ support) c",
"state_before": "case mp.mk.mk.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nl : List (Perm α)\nh : ∃ a, a ∈ l ∧ (Finset.card ∘ support) a = n\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nhld : List.Pairwise Disjoint l\n⊢ ∃ c τ, List.prod l = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n",
"tactic": "obtain ⟨c, cl, rfl⟩ := h"
},
{
"state_after": "case mp.mk.mk.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nl : List (Perm α)\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nc : Perm α\nhld : List.Pairwise Disjoint (c :: List.erase l c)\ncl : c ∈ l\n⊢ ∃ c_1 τ, List.prod l = c_1 * τ ∧ Disjoint c_1 τ ∧ IsCycle c_1 ∧ Finset.card (support c_1) = (Finset.card ∘ support) c",
"state_before": "case mp.mk.mk.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nl : List (Perm α)\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nhld : List.Pairwise Disjoint l\nc : Perm α\ncl : c ∈ l\n⊢ ∃ c_1 τ, List.prod l = c_1 * τ ∧ Disjoint c_1 τ ∧ IsCycle c_1 ∧ Finset.card (support c_1) = (Finset.card ∘ support) c",
"tactic": "rw [(List.perm_cons_erase cl).pairwise_iff Disjoint.symmetric] at hld"
},
{
"state_after": "case mp.mk.mk.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nl : List (Perm α)\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nc : Perm α\nhld : List.Pairwise Disjoint (c :: List.erase l c)\ncl : c ∈ l\n⊢ List.prod l = c * List.prod (List.erase l c)\n\ncase mp.mk.mk.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nl : List (Perm α)\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nc : Perm α\nhld : List.Pairwise Disjoint (c :: List.erase l c)\ncl : c ∈ l\n⊢ Disjoint c (List.prod (List.erase l c))",
"state_before": "case mp.mk.mk.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nl : List (Perm α)\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nc : Perm α\nhld : List.Pairwise Disjoint (c :: List.erase l c)\ncl : c ∈ l\n⊢ ∃ c_1 τ, List.prod l = c_1 * τ ∧ Disjoint c_1 τ ∧ IsCycle c_1 ∧ Finset.card (support c_1) = (Finset.card ∘ support) c",
"tactic": "refine' ⟨c, (l.erase c).prod, _, _, hlc _ cl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.mk.mk.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nl : List (Perm α)\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nc : Perm α\nhld : List.Pairwise Disjoint (c :: List.erase l c)\ncl : c ∈ l\n⊢ List.prod l = c * List.prod (List.erase l c)",
"tactic": "rw [← List.prod_cons, (List.perm_cons_erase cl).symm.prod_eq' (hld.imp Disjoint.commute)]"
},
{
"state_after": "no goals",
"state_before": "case mp.mk.mk.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nl : List (Perm α)\nx✝ : Trunc { l_1 // List.prod l_1 = List.prod l ∧ (∀ (g : Perm α), g ∈ l_1 → IsCycle g) ∧ List.Pairwise Disjoint l_1 }\nhlc : ∀ (g : Perm α), g ∈ l → IsCycle g\nc : Perm α\nhld : List.Pairwise Disjoint (c :: List.erase l c)\ncl : c ∈ l\n⊢ Disjoint c (List.prod (List.erase l c))",
"tactic": "exact disjoint_prod_right _ fun g => List.rel_of_pairwise_cons hld"
},
{
"state_after": "case mpr.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nc t : Perm α\nhd : Disjoint c t\nhc : IsCycle c\n⊢ Finset.card (support c) ∈ cycleType (c * t)",
"state_before": "case mpr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nσ : Perm α\n⊢ (∃ c τ, σ = c * τ ∧ Disjoint c τ ∧ IsCycle c ∧ Finset.card (support c) = n) → n ∈ cycleType σ",
"tactic": "rintro ⟨c, t, rfl, hd, hc, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nc t : Perm α\nhd : Disjoint c t\nhc : IsCycle c\n⊢ Finset.card (support c) ∈ cycleType (c * t)",
"tactic": "simp [hd.cycleType, hc.cycleType]"
}
] |
[
304,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Combinatorics/Colex.lean
|
Colex.le_def
|
[] |
[
101,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean
|
MeasureTheory.Measure.integral_comp_mul_left
|
[
{
"state_after": "no goals",
"state_before": "E : Type ?u.85697\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type u_1\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns : Set E\ng : ℝ → F\na : ℝ\n⊢ (∫ (x : ℝ), g (a * x)) = abs a⁻¹ • ∫ (y : ℝ), g y",
"tactic": "simp_rw [← smul_eq_mul, Measure.integral_comp_smul, FiniteDimensional.finrank_self, pow_one]"
}
] |
[
113,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.minpoly_gen
|
[
{
"state_after": "F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nα✝ : E\nK : Type ?u.813260\ninst✝¹ : Field K\ninst✝ : Algebra F K\nα : E\nh : IsIntegral F (↑(algebraMap { x // x ∈ F⟮α⟯ } E) (AdjoinSimple.gen F α))\n⊢ minpoly F (AdjoinSimple.gen F α) = minpoly F α",
"state_before": "F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nα✝ : E\nK : Type ?u.813260\ninst✝¹ : Field K\ninst✝ : Algebra F K\nα : E\nh : IsIntegral F α\n⊢ minpoly F (AdjoinSimple.gen F α) = minpoly F α",
"tactic": "rw [← AdjoinSimple.algebraMap_gen F α] at h"
},
{
"state_after": "F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nα✝ : E\nK : Type ?u.813260\ninst✝¹ : Field K\ninst✝ : Algebra F K\nα : E\nh : IsIntegral F (↑(algebraMap { x // x ∈ F⟮α⟯ } E) (AdjoinSimple.gen F α))\ninj : Function.Injective ↑(algebraMap { x // x ∈ F⟮α⟯ } E)\n⊢ minpoly F (AdjoinSimple.gen F α) = minpoly F α",
"state_before": "F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nα✝ : E\nK : Type ?u.813260\ninst✝¹ : Field K\ninst✝ : Algebra F K\nα : E\nh : IsIntegral F (↑(algebraMap { x // x ∈ F⟮α⟯ } E) (AdjoinSimple.gen F α))\n⊢ minpoly F (AdjoinSimple.gen F α) = minpoly F α",
"tactic": "have inj := (algebraMap F⟮α⟯ E).injective"
},
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nα✝ : E\nK : Type ?u.813260\ninst✝¹ : Field K\ninst✝ : Algebra F K\nα : E\nh : IsIntegral F (↑(algebraMap { x // x ∈ F⟮α⟯ } E) (AdjoinSimple.gen F α))\ninj : Function.Injective ↑(algebraMap { x // x ∈ F⟮α⟯ } E)\n⊢ minpoly F (AdjoinSimple.gen F α) = minpoly F α",
"tactic": "exact\n minpoly.eq_of_algebraMap_eq inj ((isIntegral_algebraMap_iff inj).mp h)\n (AdjoinSimple.algebraMap_gen _ _).symm"
}
] |
[
802,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
796,
1
] |
Mathlib/Order/Heyting/Hom.lean
|
HeytingHom.ext
|
[] |
[
273,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
|
Complex.cpow_one
|
[
{
"state_after": "no goals",
"state_before": "x : ℂ\nhx : x = 0\n⊢ x ^ 1 = x",
"tactic": "simp [hx, cpow_def]"
},
{
"state_after": "no goals",
"state_before": "x : ℂ\nhx : ¬x = 0\n⊢ x ^ 1 = x",
"tactic": "rw [cpow_def, if_neg (one_ne_zero : (1 : ℂ) ≠ 0), if_neg hx, mul_one, exp_log hx]"
}
] |
[
85,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
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