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 |
|---|---|---|---|---|---|---|
src/lean/Init/Data/Nat/Basic.lean
|
Nat.zero_lt_of_lt
|
[] |
[
320,
23
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
316,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.setOf_false
|
[] |
[
568,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/Topology/Covering.lean
|
IsCoveringMapOn.mk
|
[] |
[
93,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
exists_glb_Ioi
|
[] |
[
588,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.ValidFor.valid
|
[
{
"state_after": "no goals",
"state_before": "l r : List Char\n⊢ Valid { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } }",
"tactic": "simpa [List.reverseAux_eq] using Pos.Valid.mk l.reverse r"
}
] |
[
513,
74
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
512,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.coe_piCongr'
|
[] |
[
1821,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1819,
1
] |
Mathlib/Order/Interval.lean
|
Interval.coe_subset_coe
|
[] |
[
478,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.dist_le_infDist_add_diam
|
[
{
"state_after": "ι : Sort ?u.63481\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nhs : Bounded s\nhy : y ∈ s\n⊢ ENNReal.toReal (edist x y) ≤ ENNReal.toReal (infEdist x s) + ENNReal.toReal (EMetric.diam s)",
"state_before": "ι : Sort ?u.63481\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nhs : Bounded s\nhy : y ∈ s\n⊢ dist x y ≤ infDist x s + diam s",
"tactic": "rw [infDist, diam, dist_edist]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.63481\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nhs : Bounded s\nhy : y ∈ s\n⊢ ENNReal.toReal (edist x y) ≤ ENNReal.toReal (infEdist x s) + ENNReal.toReal (EMetric.diam s)",
"tactic": "exact toReal_le_add (edist_le_infEdist_add_ediam hy) (infEdist_ne_top ⟨y, hy⟩) hs.ediam_ne_top"
}
] |
[
557,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
554,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearEquiv.arrowCongr_comp
|
[
{
"state_after": "case h\nR : Type u_4\nR₁ : Type ?u.2165977\nR₂ : Type ?u.2165980\nR₃ : Type ?u.2165983\nR₄ : Type ?u.2165986\nS : Type ?u.2165989\nK : Type ?u.2165992\nK₂ : Type ?u.2165995\nM : Type u_5\nM' : Type ?u.2166001\nM₁ : Type ?u.2166004\nM₂ : Type u_6\nM₃ : Type u_7\nM₄ : Type ?u.2166013\nN✝ : Type ?u.2166016\nN₂✝ : Type ?u.2166019\nι : Type ?u.2166022\nV : Type ?u.2166025\nV₂ : Type ?u.2166028\ninst✝¹² : CommSemiring R\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₂\ninst✝⁶ : Module R M₃\nN : Type u_1\nN₂ : Type u_2\nN₃ : Type u_3\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N₃\ninst✝² : Module R N\ninst✝¹ : Module R N₂\ninst✝ : Module R N₃\ne₁ : M ≃ₗ[R] N\ne₂ : M₂ ≃ₗ[R] N₂\ne₃ : M₃ ≃ₗ[R] N₃\nf : M →ₗ[R] M₂\ng : M₂ →ₗ[R] M₃\nx✝ : N\n⊢ ↑(↑(arrowCongr e₁ e₃) (LinearMap.comp g f)) x✝ = ↑(LinearMap.comp (↑(arrowCongr e₂ e₃) g) (↑(arrowCongr e₁ e₂) f)) x✝",
"state_before": "R : Type u_4\nR₁ : Type ?u.2165977\nR₂ : Type ?u.2165980\nR₃ : Type ?u.2165983\nR₄ : Type ?u.2165986\nS : Type ?u.2165989\nK : Type ?u.2165992\nK₂ : Type ?u.2165995\nM : Type u_5\nM' : Type ?u.2166001\nM₁ : Type ?u.2166004\nM₂ : Type u_6\nM₃ : Type u_7\nM₄ : Type ?u.2166013\nN✝ : Type ?u.2166016\nN₂✝ : Type ?u.2166019\nι : Type ?u.2166022\nV : Type ?u.2166025\nV₂ : Type ?u.2166028\ninst✝¹² : CommSemiring R\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₂\ninst✝⁶ : Module R M₃\nN : Type u_1\nN₂ : Type u_2\nN₃ : Type u_3\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N₃\ninst✝² : Module R N\ninst✝¹ : Module R N₂\ninst✝ : Module R N₃\ne₁ : M ≃ₗ[R] N\ne₂ : M₂ ≃ₗ[R] N₂\ne₃ : M₃ ≃ₗ[R] N₃\nf : M →ₗ[R] M₂\ng : M₂ →ₗ[R] M₃\n⊢ ↑(arrowCongr e₁ e₃) (LinearMap.comp g f) = LinearMap.comp (↑(arrowCongr e₂ e₃) g) (↑(arrowCongr e₁ e₂) f)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_4\nR₁ : Type ?u.2165977\nR₂ : Type ?u.2165980\nR₃ : Type ?u.2165983\nR₄ : Type ?u.2165986\nS : Type ?u.2165989\nK : Type ?u.2165992\nK₂ : Type ?u.2165995\nM : Type u_5\nM' : Type ?u.2166001\nM₁ : Type ?u.2166004\nM₂ : Type u_6\nM₃ : Type u_7\nM₄ : Type ?u.2166013\nN✝ : Type ?u.2166016\nN₂✝ : Type ?u.2166019\nι : Type ?u.2166022\nV : Type ?u.2166025\nV₂ : Type ?u.2166028\ninst✝¹² : CommSemiring R\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₂\ninst✝⁶ : Module R M₃\nN : Type u_1\nN₂ : Type u_2\nN₃ : Type u_3\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N₃\ninst✝² : Module R N\ninst✝¹ : Module R N₂\ninst✝ : Module R N₃\ne₁ : M ≃ₗ[R] N\ne₂ : M₂ ≃ₗ[R] N₂\ne₃ : M₃ ≃ₗ[R] N₃\nf : M →ₗ[R] M₂\ng : M₂ →ₗ[R] M₃\nx✝ : N\n⊢ ↑(↑(arrowCongr e₁ e₃) (LinearMap.comp g f)) x✝ = ↑(LinearMap.comp (↑(arrowCongr e₂ e₃) g) (↑(arrowCongr e₁ e₂) f)) x✝",
"tactic": "simp only [symm_apply_apply, arrowCongr_apply, LinearMap.comp_apply]"
}
] |
[
2345,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2340,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.smul_linear
|
[] |
[
244,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Std/Data/List/Lemmas.lean
|
List.eraseP_cons
|
[] |
[
929,
67
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
928,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.FactorSet.prod_eq_zero_iff
|
[
{
"state_after": "case top\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : Nontrivial α\n⊢ prod ⊤ = 0 ↔ ⊤ = ⊤\n\ncase coe\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : Nontrivial α\na✝ : Multiset { a // Irreducible a }\n⊢ prod ↑a✝ = 0 ↔ ↑a✝ = ⊤",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : Nontrivial α\np : FactorSet α\n⊢ prod p = 0 ↔ p = ⊤",
"tactic": "induction p using WithTop.recTopCoe"
},
{
"state_after": "no goals",
"state_before": "case coe\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : Nontrivial α\na✝ : Multiset { a // Irreducible a }\n⊢ prod ↑a✝ = 0 ↔ ↑a✝ = ⊤",
"tactic": ". rw [prod_coe, Multiset.prod_eq_zero_iff, Multiset.mem_map, eq_false WithTop.coe_ne_top,\n iff_false_iff, not_exists]\n exact fun a => not_and_of_not_right _ a.prop.ne_zero"
},
{
"state_after": "no goals",
"state_before": "case top\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : Nontrivial α\n⊢ prod ⊤ = 0 ↔ ⊤ = ⊤",
"tactic": "simp only [iff_self_iff, eq_self_iff_true, Associates.prod_top]"
},
{
"state_after": "case coe\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : Nontrivial α\na✝ : Multiset { a // Irreducible a }\n⊢ ∀ (x : { a // Irreducible a }), ¬(x ∈ a✝ ∧ ↑x = 0)",
"state_before": "case coe\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : Nontrivial α\na✝ : Multiset { a // Irreducible a }\n⊢ prod ↑a✝ = 0 ↔ ↑a✝ = ⊤",
"tactic": "rw [prod_coe, Multiset.prod_eq_zero_iff, Multiset.mem_map, eq_false WithTop.coe_ne_top,\n iff_false_iff, not_exists]"
},
{
"state_after": "no goals",
"state_before": "case coe\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : Nontrivial α\na✝ : Multiset { a // Irreducible a }\n⊢ ∀ (x : { a // Irreducible a }), ¬(x ∈ a✝ ∧ ↑x = 0)",
"tactic": "exact fun a => not_and_of_not_right _ a.prop.ne_zero"
}
] |
[
1272,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1267,
1
] |
Mathlib/MeasureTheory/PiSystem.lean
|
isPiSystem_Ioo_mem
|
[] |
[
176,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/GroupTheory/GroupAction/Basic.lean
|
AddAction.stabilizer_vadd_eq_stabilizer_map_conj
|
[
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : AddGroup α\ninst✝ : AddAction α β\ng : α\nx : β\nh : α\n⊢ h ∈ stabilizer α (g +ᵥ x) ↔ h ∈ AddSubgroup.map (AddEquiv.toAddMonoidHom (↑AddAut.conj g)) (stabilizer α x)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : AddGroup α\ninst✝ : AddAction α β\ng : α\nx : β\n⊢ stabilizer α (g +ᵥ x) = AddSubgroup.map (AddEquiv.toAddMonoidHom (↑AddAut.conj g)) (stabilizer α x)",
"tactic": "ext h"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : AddGroup α\ninst✝ : AddAction α β\ng : α\nx : β\nh : α\n⊢ h ∈ stabilizer α (g +ᵥ x) ↔ h ∈ AddSubgroup.map (AddEquiv.toAddMonoidHom (↑AddAut.conj g)) (stabilizer α x)",
"tactic": "rw [mem_stabilizer_iff, ← vadd_left_cancel_iff (-g), vadd_vadd, vadd_vadd, vadd_vadd,\n add_left_neg, zero_vadd, ← mem_stabilizer_iff, AddSubgroup.mem_map_equiv,\n AddAut.conj_symm_apply]"
}
] |
[
448,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.Nonempty.biUnion
|
[] |
[
3669,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3667,
1
] |
Mathlib/GroupTheory/GroupAction/SubMulAction.lean
|
SubMulAction.mem_carrier
|
[] |
[
130,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean
|
fixedPoints_subgroup_iSup
|
[] |
[
164,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
Class.mem_def
|
[] |
[
1501,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1500,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
strictMonoOn_comp_ofDual_iff
|
[] |
[
202,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.aevalTower_comp_algebraMap
|
[] |
[
421,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval₂_eta
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\n⊢ eval₂ C X p = p",
"tactic": "apply MvPolynomial.induction_on p <;>\n simp (config := { contextual := true }) [eval₂_add, eval₂_mul]"
}
] |
[
1085,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1083,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.mk_eq
|
[] |
[
520,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.update_comp_eq_of_injective'
|
[] |
[
632,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
630,
1
] |
Mathlib/Algebra/Homology/Homotopy.lean
|
Homotopy.prevD_succ_cochainComplex
|
[
{
"state_after": "ι : Type ?u.373772\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\ni : ℕ\n⊢ f (i + 1) (ComplexShape.prev (ComplexShape.up ℕ) (i + 1)) ≫\n d Q (ComplexShape.prev (ComplexShape.up ℕ) (i + 1)) (i + 1) =\n f (i + 1) i ≫ d Q i (i + 1)",
"state_before": "ι : Type ?u.373772\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\ni : ℕ\n⊢ ↑(prevD (i + 1)) f = f (i + 1) i ≫ d Q i (i + 1)",
"tactic": "dsimp [prevD]"
},
{
"state_after": "ι : Type ?u.373772\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\ni : ℕ\nthis : ComplexShape.prev (ComplexShape.up ℕ) (i + 1) = i\n⊢ f (i + 1) (ComplexShape.prev (ComplexShape.up ℕ) (i + 1)) ≫\n d Q (ComplexShape.prev (ComplexShape.up ℕ) (i + 1)) (i + 1) =\n f (i + 1) i ≫ d Q i (i + 1)",
"state_before": "ι : Type ?u.373772\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\ni : ℕ\n⊢ f (i + 1) (ComplexShape.prev (ComplexShape.up ℕ) (i + 1)) ≫\n d Q (ComplexShape.prev (ComplexShape.up ℕ) (i + 1)) (i + 1) =\n f (i + 1) i ≫ d Q i (i + 1)",
"tactic": "have : (ComplexShape.up ℕ).prev (i + 1) = i := CochainComplex.prev_nat_succ i"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.373772\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni✝ : ι\nP Q : CochainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\ni : ℕ\nthis : ComplexShape.prev (ComplexShape.up ℕ) (i + 1) = i\n⊢ f (i + 1) (ComplexShape.prev (ComplexShape.up ℕ) (i + 1)) ≫\n d Q (ComplexShape.prev (ComplexShape.up ℕ) (i + 1)) (i + 1) =\n f (i + 1) i ≫ d Q i (i + 1)",
"tactic": "congr 2"
}
] |
[
599,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
595,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.map_eq_neg_one_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.77751\nα : Type ?u.77754\nβ : Type ?u.77757\nR : Type u_2\nS : Type u_1\nS' : Type ?u.77766\ninst✝¹ : NonAssocRing R\ninst✝ : NonAssocRing S\nf : R ≃+* S\nx✝ y x : R\n⊢ ↑f x = -1 ↔ x = -1",
"tactic": "rw [← neg_eq_iff_eq_neg, ← neg_eq_iff_eq_neg, ← map_neg, RingEquiv.map_eq_one_iff]"
}
] |
[
588,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.InjOn.preimage_image_inter
|
[] |
[
719,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
718,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
CategoryTheory.Limits.kernelIsoOfEq_hom_comp_ι
|
[
{
"state_after": "case refl\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nf : X ⟶ Y\ninst✝¹ inst✝ : HasKernel f\n⊢ (kernelIsoOfEq (_ : f = f)).hom ≫ kernel.ι f = kernel.ι f",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nf g : X ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasKernel g\nh : f = g\n⊢ (kernelIsoOfEq h).hom ≫ kernel.ι g = kernel.ι f",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nX Y : C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nf : X ⟶ Y\ninst✝¹ inst✝ : HasKernel f\n⊢ (kernelIsoOfEq (_ : f = f)).hom ≫ kernel.ι f = kernel.ι f",
"tactic": "simp"
}
] |
[
359,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
lt_mul_of_one_lt_left'
|
[] |
[
460,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Order/Antichain.lean
|
isAntichain_and_greatest_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.11422\nr r₁ r₂ : α → α → Prop\nr' : β → β → Prop\nt : Set α\na b : α\ninst✝ : Preorder α\n⊢ IsAntichain (fun x x_1 => x ≤ x_1) {a} ∧ IsGreatest {a} a",
"state_before": "α : Type u_1\nβ : Type ?u.11422\nr r₁ r₂ : α → α → Prop\nr' : β → β → Prop\ns t : Set α\na b : α\ninst✝ : Preorder α\n⊢ s = {a} → IsAntichain (fun x x_1 => x ≤ x_1) s ∧ IsGreatest s a",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11422\nr r₁ r₂ : α → α → Prop\nr' : β → β → Prop\nt : Set α\na b : α\ninst✝ : Preorder α\n⊢ IsAntichain (fun x x_1 => x ≤ x_1) {a} ∧ IsGreatest {a} a",
"tactic": "exact ⟨isAntichain_singleton _ _, isGreatest_singleton⟩"
}
] |
[
233,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.IsSeparating.isDetecting
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\n⊢ IsIso f",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\n⊢ IsDetecting 𝒢",
"tactic": "intro X Y f hf"
},
{
"state_after": "case refine'_1\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Z✝ ⟶ X\nhgh : g ≫ f = h ≫ f\nG : C\nhG : G ∈ 𝒢\ni : G ⟶ Z✝\n⊢ i ≫ g = i ≫ h\n\ncase refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Y ⟶ Z✝\nhgh : f ≫ g = f ≫ h\n⊢ g = h",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\n⊢ IsIso f",
"tactic": "refine'\n (isIso_iff_mono_and_epi _).2 ⟨⟨fun g h hgh => h𝒢 _ _ fun G hG i => _⟩, ⟨fun g h hgh => _⟩⟩"
},
{
"state_after": "case refine'_1.intro.intro\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Z✝ ⟶ X\nhgh : g ≫ f = h ≫ f\nG : C\nhG : G ∈ 𝒢\ni : G ⟶ Z✝\nt : G ⟶ X\nht : ∀ (y : G ⟶ X), (fun h' => h' ≫ f = i ≫ g ≫ f) y → y = t\n⊢ i ≫ g = i ≫ h",
"state_before": "case refine'_1\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Z✝ ⟶ X\nhgh : g ≫ f = h ≫ f\nG : C\nhG : G ∈ 𝒢\ni : G ⟶ Z✝\n⊢ i ≫ g = i ≫ h",
"tactic": "obtain ⟨t, -, ht⟩ := hf G hG (i ≫ g ≫ f)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Z✝ ⟶ X\nhgh : g ≫ f = h ≫ f\nG : C\nhG : G ∈ 𝒢\ni : G ⟶ Z✝\nt : G ⟶ X\nht : ∀ (y : G ⟶ X), (fun h' => h' ≫ f = i ≫ g ≫ f) y → y = t\n⊢ i ≫ g = i ≫ h",
"tactic": "rw [ht (i ≫ g) (Category.assoc _ _ _), ht (i ≫ h) (hgh.symm ▸ Category.assoc _ _ _)]"
},
{
"state_after": "case refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Y ⟶ Z✝\nhgh : f ≫ g = f ≫ h\nG : C\nhG : G ∈ 𝒢\ni : G ⟶ Y\n⊢ i ≫ g = i ≫ h",
"state_before": "case refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Y ⟶ Z✝\nhgh : f ≫ g = f ≫ h\n⊢ g = h",
"tactic": "refine' h𝒢 _ _ fun G hG i => _"
},
{
"state_after": "case refine'_2.intro.intro\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Y ⟶ Z✝\nhgh : f ≫ g = f ≫ h\nG : C\nhG : G ∈ 𝒢\nt : G ⟶ X\n⊢ (t ≫ f) ≫ g = (t ≫ f) ≫ h",
"state_before": "case refine'_2\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Y ⟶ Z✝\nhgh : f ≫ g = f ≫ h\nG : C\nhG : G ∈ 𝒢\ni : G ⟶ Y\n⊢ i ≫ g = i ≫ h",
"tactic": "obtain ⟨t, rfl, -⟩ := hf G hG i"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\nh𝒢 : IsSeparating 𝒢\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), G ∈ 𝒢 → ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h\nZ✝ : C\ng h : Y ⟶ Z✝\nhgh : f ≫ g = f ≫ h\nG : C\nhG : G ∈ 𝒢\nt : G ⟶ X\n⊢ (t ≫ f) ≫ g = (t ≫ f) ≫ h",
"tactic": "rw [Category.assoc, hgh, Category.assoc]"
}
] |
[
177,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.singleton_subset_singleton
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\n⊢ {a} ⊆ {b} ↔ a = b",
"tactic": "simp"
}
] |
[
1317,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1317,
1
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
|
HasFDerivWithinAt.exp
|
[] |
[
279,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/LinearAlgebra/Matrix/Symmetric.lean
|
Matrix.isSymm_fromBlocks_iff
|
[] |
[
151,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Analysis/SpecialFunctions/Arsinh.lean
|
Real.sinh_bijective
|
[] |
[
97,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
Ideal.le_of_pow_le_prime
|
[
{
"state_after": "case pos\nR : Type u_1\nA : Type ?u.1186416\nK : Type ?u.1186419\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI P : Ideal R\nhP : IsPrime P\nn : ℕ\nh : I ^ n ≤ P\nhP0 : P = ⊥\n⊢ I ≤ P\n\ncase neg\nR : Type u_1\nA : Type ?u.1186416\nK : Type ?u.1186419\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI P : Ideal R\nhP : IsPrime P\nn : ℕ\nh : I ^ n ≤ P\nhP0 : ¬P = ⊥\n⊢ I ≤ P",
"state_before": "R : Type u_1\nA : Type ?u.1186416\nK : Type ?u.1186419\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI P : Ideal R\nhP : IsPrime P\nn : ℕ\nh : I ^ n ≤ P\n⊢ I ≤ P",
"tactic": "by_cases hP0 : P = ⊥"
},
{
"state_after": "case neg\nR : Type u_1\nA : Type ?u.1186416\nK : Type ?u.1186419\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI P : Ideal R\nhP : IsPrime P\nn : ℕ\nh : P ∣ I ^ n\nhP0 : ¬P = ⊥\n⊢ P ∣ I",
"state_before": "case neg\nR : Type u_1\nA : Type ?u.1186416\nK : Type ?u.1186419\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI P : Ideal R\nhP : IsPrime P\nn : ℕ\nh : I ^ n ≤ P\nhP0 : ¬P = ⊥\n⊢ I ≤ P",
"tactic": "rw [← Ideal.dvd_iff_le] at h ⊢"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\nA : Type ?u.1186416\nK : Type ?u.1186419\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI P : Ideal R\nhP : IsPrime P\nn : ℕ\nh : P ∣ I ^ n\nhP0 : ¬P = ⊥\n⊢ P ∣ I",
"tactic": "exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_of_dvd_pow h"
},
{
"state_after": "case pos\nR : Type u_1\nA : Type ?u.1186416\nK : Type ?u.1186419\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI P : Ideal R\nhP : IsPrime P\nn : ℕ\nhP0 : P = ⊥\nh : I ^ n = ⊥\n⊢ I = ⊥",
"state_before": "case pos\nR : Type u_1\nA : Type ?u.1186416\nK : Type ?u.1186419\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI P : Ideal R\nhP : IsPrime P\nn : ℕ\nh : I ^ n ≤ P\nhP0 : P = ⊥\n⊢ I ≤ P",
"tactic": "simp only [hP0, le_bot_iff] at h ⊢"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\nA : Type ?u.1186416\nK : Type ?u.1186419\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nI P : Ideal R\nhP : IsPrime P\nn : ℕ\nhP0 : P = ⊥\nh : I ^ n = ⊥\n⊢ I = ⊥",
"tactic": "exact pow_eq_zero h"
}
] |
[
1256,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1250,
1
] |
Mathlib/Order/Lattice.lean
|
Antitone.map_sup_le
|
[] |
[
1190,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1188,
1
] |
Std/Data/List/Basic.lean
|
List.product_eq_productTR
|
[
{
"state_after": "case h.h.h.h\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ product l₁ l₂ = productTR l₁ l₂",
"state_before": "⊢ @product = @productTR",
"tactic": "funext α β l₁ l₂"
},
{
"state_after": "case h.h.h.h\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ (List.bind l₁ fun a => map (Prod.mk a) l₂) =\n (foldl (fun acc a => foldl (fun acc b => Array.push acc (a, b)) acc l₂) #[] l₁).data",
"state_before": "case h.h.h.h\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ product l₁ l₂ = productTR l₁ l₂",
"tactic": "simp [product, productTR]"
},
{
"state_after": "case h.h.h.h\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ (List.bind l₁ fun a => map (Prod.mk a) l₂) = #[].data ++ List.bind l₁ ?h.h.h.h.G\n\ncase h.h.h.h.G\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ α → List (α × β)\n\ncase h.h.h.h.H\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ ∀ (acc : Array (α × β)) (a : α), (foldl (fun acc b => Array.push acc (a, b)) acc l₂).data = acc.data ++ ?h.h.h.h.G a",
"state_before": "case h.h.h.h\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ (List.bind l₁ fun a => map (Prod.mk a) l₂) =\n (foldl (fun acc a => foldl (fun acc b => Array.push acc (a, b)) acc l₂) #[] l₁).data",
"tactic": "rw [Array.foldl_data_eq_bind]"
},
{
"state_after": "case h.h.h.h.H\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ ∀ (acc : Array (α × β)) (a : α),\n (foldl (fun acc b => Array.push acc (a, b)) acc l₂).data = acc.data ++ map (Prod.mk a) l₂",
"state_before": "case h.h.h.h\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ (List.bind l₁ fun a => map (Prod.mk a) l₂) = #[].data ++ List.bind l₁ ?h.h.h.h.G\n\ncase h.h.h.h.G\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ α → List (α × β)\n\ncase h.h.h.h.H\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ ∀ (acc : Array (α × β)) (a : α), (foldl (fun acc b => Array.push acc (a, b)) acc l₂).data = acc.data ++ ?h.h.h.h.G a",
"tactic": "rfl"
},
{
"state_after": "case h.h.h.h.H\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\nacc✝ : Array (α × β)\na✝ : α\n⊢ (foldl (fun acc b => Array.push acc (a✝, b)) acc✝ l₂).data = acc✝.data ++ map (Prod.mk a✝) l₂",
"state_before": "case h.h.h.h.H\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\n⊢ ∀ (acc : Array (α × β)) (a : α),\n (foldl (fun acc b => Array.push acc (a, b)) acc l₂).data = acc.data ++ map (Prod.mk a) l₂",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.h.H\nα : Type u_2\nβ : Type u_1\nl₁ : List α\nl₂ : List β\nacc✝ : Array (α × β)\na✝ : α\n⊢ (foldl (fun acc b => Array.push acc (a✝, b)) acc✝ l₂).data = acc✝.data ++ map (Prod.mk a✝) l₂",
"tactic": "apply Array.foldl_data_eq_map"
}
] |
[
989,
40
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
986,
10
] |
Mathlib/Topology/Sheaves/Presheaf.lean
|
TopCat.Presheaf.Pushforward.id_inv_app'
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nX : TopCat\nℱ : Presheaf C X\nU : Set ↑X\np : IsOpen U\n⊢ ((Functor.leftUnitor ℱ).inv ≫ whiskerRight (NatTrans.op (Opens.mapId X).hom) ℱ).app\n { carrier := U, is_open' := p }.op =\n ℱ.map (𝟙 { carrier := U, is_open' := p }.op)",
"state_before": "C : Type u\ninst✝ : Category C\nX : TopCat\nℱ : Presheaf C X\nU : Set ↑X\np : IsOpen U\n⊢ (id ℱ).inv.app { carrier := U, is_open' := p }.op = ℱ.map (𝟙 { carrier := U, is_open' := p }.op)",
"tactic": "dsimp [id]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX : TopCat\nℱ : Presheaf C X\nU : Set ↑X\np : IsOpen U\n⊢ ((Functor.leftUnitor ℱ).inv ≫ whiskerRight (NatTrans.op (Opens.mapId X).hom) ℱ).app\n { carrier := U, is_open' := p }.op =\n ℱ.map (𝟙 { carrier := U, is_open' := p }.op)",
"tactic": "simp [CategoryStruct.comp]"
}
] |
[
239,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Std/Logic.lean
|
or_iff_right_of_imp
|
[] |
[
289,
79
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
289,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.toList_nodup
|
[] |
[
265,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Data/Set/Function.lean
|
Function.Semiconj.surjOn_image
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.111397\nι : Sort ?u.111400\nπ : α → Type ?u.111405\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : SurjOn fa s t\nx : α\nhxt : x ∈ t\n⊢ f x ∈ fb '' (f '' s)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.111397\nι : Sort ?u.111400\nπ : α → Type ?u.111405\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : SurjOn fa s t\n⊢ SurjOn fb (f '' s) (f '' t)",
"tactic": "rintro y ⟨x, hxt, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.111397\nι : Sort ?u.111400\nπ : α → Type ?u.111405\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : SurjOn fa s t\nx : α\nhxs : x ∈ s\nhxt : fa x ∈ t\n⊢ f (fa x) ∈ fb '' (f '' s)",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.111397\nι : Sort ?u.111400\nπ : α → Type ?u.111405\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : SurjOn fa s t\nx : α\nhxt : x ∈ t\n⊢ f x ∈ fb '' (f '' s)",
"tactic": "rcases ha hxt with ⟨x, hxs, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.111397\nι : Sort ?u.111400\nπ : α → Type ?u.111405\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : SurjOn fa s t\nx : α\nhxs : x ∈ s\nhxt : fa x ∈ t\n⊢ fb (f x) ∈ fb '' (f '' s)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.111397\nι : Sort ?u.111400\nπ : α → Type ?u.111405\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : SurjOn fa s t\nx : α\nhxs : x ∈ s\nhxt : fa x ∈ t\n⊢ f (fa x) ∈ fb '' (f '' s)",
"tactic": "rw [h x]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.111397\nι : Sort ?u.111400\nπ : α → Type ?u.111405\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns t : Set α\nh : Semiconj f fa fb\nha : SurjOn fa s t\nx : α\nhxs : x ∈ s\nhxt : fa x ∈ t\n⊢ fb (f x) ∈ fb '' (f '' s)",
"tactic": "exact mem_image_of_mem _ (mem_image_of_mem _ hxs)"
}
] |
[
1630,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1626,
1
] |
Mathlib/Topology/Spectral/Hom.lean
|
SpectralMap.toFun_eq_coe
|
[] |
[
129,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Topology/LocallyConstant/Algebra.lean
|
LocallyConstant.mul_apply
|
[] |
[
68,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Data/Analysis/Filter.lean
|
Filter.Realizer.ne_bot_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.55845\nσ : Type ?u.55848\nτ : Type ?u.55851\nf : Filter α\nF : Realizer f\n⊢ (∃ x, ¬Set.Nonempty (CFilter.f F.F x)) ↔ ∀ (b : Realizer.bot.σ), ∃ a, CFilter.f F.F a ≤ CFilter.f Realizer.bot.F b",
"state_before": "α : Type u_1\nβ : Type ?u.55845\nσ : Type ?u.55848\nτ : Type ?u.55851\nf : Filter α\nF : Realizer f\n⊢ f ≠ ⊥ ↔ ∀ (a : F.σ), Set.Nonempty (CFilter.f F.F a)",
"tactic": "rw [not_iff_comm, ← le_bot_iff, F.le_iff Realizer.bot, not_forall]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.55845\nσ : Type ?u.55848\nτ : Type ?u.55851\nf : Filter α\nF : Realizer f\n⊢ (∃ x, CFilter.f F.F x = ∅) ↔ ∀ (b : Realizer.bot.σ), ∃ a, CFilter.f F.F a ≤ CFilter.f Realizer.bot.F b",
"state_before": "α : Type u_1\nβ : Type ?u.55845\nσ : Type ?u.55848\nτ : Type ?u.55851\nf : Filter α\nF : Realizer f\n⊢ (∃ x, ¬Set.Nonempty (CFilter.f F.F x)) ↔ ∀ (b : Realizer.bot.σ), ∃ a, CFilter.f F.F a ≤ CFilter.f Realizer.bot.F b",
"tactic": "simp only [Set.not_nonempty_iff_eq_empty]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.55845\nσ : Type ?u.55848\nτ : Type ?u.55851\nf : Filter α\nF : Realizer f\n⊢ (∃ x, CFilter.f F.F x = ∅) ↔ ∀ (b : Realizer.bot.σ), ∃ a, CFilter.f F.F a ≤ CFilter.f Realizer.bot.F b",
"tactic": "exact ⟨fun ⟨x, e⟩ _ ↦ ⟨x, le_of_eq e⟩, fun h ↦\n let ⟨x, h⟩ := h ()\n ⟨x, le_bot_iff.1 h⟩⟩"
}
] |
[
355,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.splits_of_splits_gcd_right
|
[] |
[
266,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Data/Nat/WithBot.lean
|
Nat.WithBot.add_eq_three_iff
|
[
{
"state_after": "case none.none\n\n⊢ none + none = 3 ↔ none = 0 ∧ none = 3 ∨ none = 1 ∧ none = 2 ∨ none = 2 ∧ none = 1 ∨ none = 3 ∧ none = 0\n\ncase none.some\nval✝ : ℕ\n⊢ none + some val✝ = 3 ↔\n none = 0 ∧ some val✝ = 3 ∨ none = 1 ∧ some val✝ = 2 ∨ none = 2 ∧ some val✝ = 1 ∨ none = 3 ∧ some val✝ = 0\n\ncase some.none\nval✝ : ℕ\n⊢ some val✝ + none = 3 ↔\n some val✝ = 0 ∧ none = 3 ∨ some val✝ = 1 ∧ none = 2 ∨ some val✝ = 2 ∧ none = 1 ∨ some val✝ = 3 ∧ none = 0\n\ncase some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 3 ↔\n some val✝¹ = 0 ∧ some val✝ = 3 ∨\n some val✝¹ = 1 ∧ some val✝ = 2 ∨ some val✝¹ = 2 ∧ some val✝ = 1 ∨ some val✝¹ = 3 ∧ some val✝ = 0",
"state_before": "n m : WithBot ℕ\n⊢ n + m = 3 ↔ n = 0 ∧ m = 3 ∨ n = 1 ∧ m = 2 ∨ n = 2 ∧ m = 1 ∨ n = 3 ∧ m = 0",
"tactic": "rcases n, m with ⟨_ | _, _ | _⟩"
},
{
"state_after": "case some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 3 ↔\n some val✝¹ = 0 ∧ some val✝ = 3 ∨\n some val✝¹ = 1 ∧ some val✝ = 2 ∨ some val✝¹ = 2 ∧ some val✝ = 1 ∨ some val✝¹ = 3 ∧ some val✝ = 0",
"state_before": "case none.none\n\n⊢ none + none = 3 ↔ none = 0 ∧ none = 3 ∨ none = 1 ∧ none = 2 ∨ none = 2 ∧ none = 1 ∨ none = 3 ∧ none = 0\n\ncase none.some\nval✝ : ℕ\n⊢ none + some val✝ = 3 ↔\n none = 0 ∧ some val✝ = 3 ∨ none = 1 ∧ some val✝ = 2 ∨ none = 2 ∧ some val✝ = 1 ∨ none = 3 ∧ some val✝ = 0\n\ncase some.none\nval✝ : ℕ\n⊢ some val✝ + none = 3 ↔\n some val✝ = 0 ∧ none = 3 ∨ some val✝ = 1 ∧ none = 2 ∨ some val✝ = 2 ∧ none = 1 ∨ some val✝ = 3 ∧ none = 0\n\ncase some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 3 ↔\n some val✝¹ = 0 ∧ some val✝ = 3 ∨\n some val✝¹ = 1 ∧ some val✝ = 2 ∨ some val✝¹ = 2 ∧ some val✝ = 1 ∨ some val✝¹ = 3 ∧ some val✝ = 0",
"tactic": "any_goals refine' ⟨fun h => Option.noConfusion h, fun h => _⟩; aesop"
},
{
"state_after": "case some.some\nval✝¹ val✝ : ℕ\n⊢ (fun x x_1 => x + x_1) val✝¹ val✝ = ↑3 ↔\n val✝¹ = 0 ∧ val✝ = ↑3 ∨ val✝¹ = 1 ∧ val✝ = ↑2 ∨ val✝¹ = ↑2 ∧ val✝ = 1 ∨ val✝¹ = ↑3 ∧ val✝ = 0",
"state_before": "case some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 3 ↔\n some val✝¹ = 0 ∧ some val✝ = 3 ∨\n some val✝¹ = 1 ∧ some val✝ = 2 ∨ some val✝¹ = 2 ∧ some val✝ = 1 ∨ some val✝¹ = 3 ∧ some val✝ = 0",
"tactic": "repeat' erw [WithBot.coe_eq_coe]"
},
{
"state_after": "no goals",
"state_before": "case some.some\nval✝¹ val✝ : ℕ\n⊢ (fun x x_1 => x + x_1) val✝¹ val✝ = ↑3 ↔\n val✝¹ = 0 ∧ val✝ = ↑3 ∨ val✝¹ = 1 ∧ val✝ = ↑2 ∨ val✝¹ = ↑2 ∧ val✝ = 1 ∨ val✝¹ = ↑3 ∧ val✝ = 0",
"tactic": "exact Nat.add_eq_three_iff"
},
{
"state_after": "case some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 3 ↔\n some val✝¹ = 0 ∧ some val✝ = 3 ∨\n some val✝¹ = 1 ∧ some val✝ = 2 ∨ some val✝¹ = 2 ∧ some val✝ = 1 ∨ some val✝¹ = 3 ∧ some val✝ = 0",
"state_before": "case some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 3 ↔\n some val✝¹ = 0 ∧ some val✝ = 3 ∨\n some val✝¹ = 1 ∧ some val✝ = 2 ∨ some val✝¹ = 2 ∧ some val✝ = 1 ∨ some val✝¹ = 3 ∧ some val✝ = 0",
"tactic": "refine' ⟨fun h => Option.noConfusion h, fun h => _⟩"
},
{
"state_after": "no goals",
"state_before": "case some.none\nval✝ : ℕ\nh : some val✝ = 0 ∧ none = 3 ∨ some val✝ = 1 ∧ none = 2 ∨ some val✝ = 2 ∧ none = 1 ∨ some val✝ = 3 ∧ none = 0\n⊢ some val✝ + none = 3",
"tactic": "aesop"
},
{
"state_after": "case some.some\nval✝¹ val✝ : ℕ\n⊢ (fun x x_1 => x + x_1) val✝¹ val✝ = ↑3 ↔\n val✝¹ = 0 ∧ val✝ = ↑3 ∨ val✝¹ = 1 ∧ val✝ = ↑2 ∨ val✝¹ = ↑2 ∧ val✝ = 1 ∨ val✝¹ = ↑3 ∧ val✝ = 0",
"state_before": "case some.some\nval✝¹ val✝ : ℕ\n⊢ (fun x x_1 => x + x_1) val✝¹ val✝ = ↑3 ↔\n val✝¹ = 0 ∧ val✝ = ↑3 ∨ val✝¹ = 1 ∧ val✝ = ↑2 ∨ val✝¹ = ↑2 ∧ val✝ = 1 ∨ val✝¹ = ↑3 ∧ val✝ = 0",
"tactic": "erw [WithBot.coe_eq_coe]"
}
] |
[
54,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
mem_image_univ_iff_mem_range
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.141366\nβ✝ : Type ?u.141369\nγ : Type ?u.141372\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : DecidableEq β\nf : α → β\nb : β\n⊢ b ∈ image f univ ↔ b ∈ Set.range f",
"tactic": "simp"
}
] |
[
1091,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1090,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
le_compl_comm
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.157568\nα : Type u_1\nβ : Type ?u.157574\ninst✝ : HeytingAlgebra α\na b c : α\n⊢ a ≤ bᶜ ↔ b ≤ aᶜ",
"tactic": "rw [le_compl_iff_disjoint_right, le_compl_iff_disjoint_left]"
}
] |
[
828,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
827,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Convex.mapsTo_lineMap
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.219178\nβ : Type ?u.219181\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t : Set E\nh : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : y ∈ s\n⊢ MapsTo (↑(AffineMap.lineMap x y)) (Icc 0 1) s",
"tactic": "simpa only [mapsTo', segment_eq_image_lineMap] using h.segment_subset hx hy"
}
] |
[
477,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Equiv.surjective
|
[] |
[
853,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.map_finsupp_prod
|
[] |
[
766,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
8
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean
|
CategoryTheory.Bicategory.associator_inv_naturality_right
|
[
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh h' : c ⟶ d\nη : h ⟶ h'\n⊢ f ◁ g ◁ η ≫ (α_ f g h').inv = (α_ f g h).inv ≫ (f ≫ g) ◁ η",
"tactic": "simp"
}
] |
[
380,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
inv_lt_div_iff_lt_mul'
|
[] |
[
996,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
995,
1
] |
Mathlib/CategoryTheory/Subobject/FactorThru.lean
|
CategoryTheory.Subobject.factorThru_right
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nX✝ Y✝ Z✝ : C\nD : Type u₂\ninst✝ : Category D\nX Y Z : C\nP : Subobject Z\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Factors P g\n⊢ (f ≫ factorThru P g h) ≫ arrow P = factorThru P (f ≫ g) (_ : Factors P (f ≫ g)) ≫ arrow P",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nX✝ Y✝ Z✝ : C\nD : Type u₂\ninst✝ : Category D\nX Y Z : C\nP : Subobject Z\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Factors P g\n⊢ f ≫ factorThru P g h = factorThru P (f ≫ g) (_ : Factors P (f ≫ g))",
"tactic": "apply (cancel_mono P.arrow).mp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nX✝ Y✝ Z✝ : C\nD : Type u₂\ninst✝ : Category D\nX Y Z : C\nP : Subobject Z\nf : X ⟶ Y\ng : Y ⟶ Z\nh : Factors P g\n⊢ (f ≫ factorThru P g h) ≫ arrow P = factorThru P (f ≫ g) (_ : Factors P (f ≫ g)) ≫ arrow P",
"tactic": "simp"
}
] |
[
164,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.op_norm_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2592188\nE : Type u_3\nEₗ : Type ?u.2592194\nF : Type u_4\nFₗ : Type ?u.2592200\nG : Type ?u.2592203\nGₗ : Type ?u.2592206\n𝓕 : Type ?u.2592209\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : NormedAddCommGroup Fₗ\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NontriviallyNormedField 𝕜₂\ninst✝⁵ : NontriviallyNormedField 𝕜₃\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\ninst✝² : NormedSpace 𝕜₃ G\ninst✝¹ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx✝ y z : E\ninst✝ : RingHomIsometric σ₁₂\nhn : ‖f‖ = 0\nx : E\n⊢ ‖f‖ * ‖x‖ = 0",
"tactic": "rw [hn, MulZeroClass.zero_mul]"
},
{
"state_after": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2592188\nE : Type u_3\nEₗ : Type ?u.2592194\nF : Type u_4\nFₗ : Type ?u.2592200\nG : Type ?u.2592203\nGₗ : Type ?u.2592206\n𝓕 : Type ?u.2592209\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : NormedAddCommGroup Fₗ\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NontriviallyNormedField 𝕜₂\ninst✝⁵ : NontriviallyNormedField 𝕜₃\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\ninst✝² : NormedSpace 𝕜₃ G\ninst✝¹ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ng : E →SL[σ₁₂] F\nx y z : E\ninst✝ : RingHomIsometric σ₁₂\n⊢ ‖0‖ = 0",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2592188\nE : Type u_3\nEₗ : Type ?u.2592194\nF : Type u_4\nFₗ : Type ?u.2592200\nG : Type ?u.2592203\nGₗ : Type ?u.2592206\n𝓕 : Type ?u.2592209\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : NormedAddCommGroup Fₗ\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NontriviallyNormedField 𝕜₂\ninst✝⁵ : NontriviallyNormedField 𝕜₃\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\ninst✝² : NormedSpace 𝕜₃ G\ninst✝¹ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf g : E →SL[σ₁₂] F\nx y z : E\ninst✝ : RingHomIsometric σ₁₂\n⊢ f = 0 → ‖f‖ = 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.2592188\nE : Type u_3\nEₗ : Type ?u.2592194\nF : Type u_4\nFₗ : Type ?u.2592200\nG : Type ?u.2592203\nGₗ : Type ?u.2592206\n𝓕 : Type ?u.2592209\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedAddCommGroup G\ninst✝⁸ : NormedAddCommGroup Fₗ\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NontriviallyNormedField 𝕜₂\ninst✝⁵ : NontriviallyNormedField 𝕜₃\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\ninst✝² : NormedSpace 𝕜₃ G\ninst✝¹ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\ng : E →SL[σ₁₂] F\nx y z : E\ninst✝ : RingHomIsometric σ₁₂\n⊢ ‖0‖ = 0",
"tactic": "exact op_norm_zero"
}
] |
[
1472,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1464,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean
|
CategoryTheory.Bicategory.pentagon_inv
|
[
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\ni : d ⟶ e\n⊢ inv (f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i) =\n inv ((α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv)",
"tactic": "simp"
}
] |
[
260,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalExtrOn.comp_mono
|
[] |
[
258,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
8
] |
Mathlib/MeasureTheory/Covering/VitaliFamily.lean
|
VitaliFamily.FineSubfamilyOn.covering_mem
|
[] |
[
147,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.BinaryFan.π_app_left
|
[] |
[
218,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
IndexedPartition.equivQuotient_index_apply
|
[] |
[
435,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Combinatorics/SetFamily/Shadow.lean
|
Finset.mem_shadow_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns t : Finset α\na : α\nk r : ℕ\n⊢ s ∈ (∂ ) 𝒜 ↔ ∃ t, t ∈ 𝒜 ∧ ∃ a, a ∈ t ∧ erase t a = s",
"tactic": "simp only [shadow, mem_sup, mem_image]"
}
] |
[
93,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.extend_apply'
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort u_2\nγ : Sort u_3\nf : α → β\ng : α → γ\ne' : β → γ\nb : β\nhb : ¬∃ a, f a = b\n⊢ extend f g e' b = e' b",
"tactic": "simp [Function.extend_def, hb]"
}
] |
[
734,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
732,
1
] |
Mathlib/GroupTheory/Coset.lean
|
orbit_subgroup_eq_rightCoset
|
[] |
[
236,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.exists_list_of_mem_closure
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh✝ : x✝ ∈ closure s\nx : R\nhx : x ∈ ↑(Submonoid.closure s)\nl : List R\nhl : ∀ (y : R), y ∈ l → y ∈ s\nh : List.prod l = x\n⊢ ∀ (y : R), y ∈ l → y ∈ s ∨ y = -1",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh✝ : x✝ ∈ closure s\nx : R\nhx : x ∈ ↑(Submonoid.closure s)\nl : List R\nhl : ∀ (y : R), y ∈ l → y ∈ s\nh : List.prod l = x\n⊢ (∀ (t : List R), t ∈ [l] → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1) ∧ List.sum (List.map List.prod [l]) = x",
"tactic": "simp [h]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh✝ : x✝ ∈ closure s\nx : R\nhx : x ∈ ↑(Submonoid.closure s)\nl : List R\nhl : ∀ (y : R), y ∈ l → y ∈ s\nh : List.prod l = x\n⊢ ∀ (y : R), y ∈ l → y ∈ s ∨ y = -1",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh✝ : x✝ ∈ closure s\nx : R\nhx : x ∈ ↑(Submonoid.closure s)\nl : List R\nhl : ∀ (y : R), y ∈ l → y ∈ s\nh : List.prod l = x\n⊢ ∀ (y : R), y ∈ l → y ∈ s ∨ y = -1",
"tactic": "clear_aux_decl"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝ : R\nh✝ : x✝ ∈ closure s\nx : R\nhx : x ∈ ↑(Submonoid.closure s)\nl : List R\nhl : ∀ (y : R), y ∈ l → y ∈ s\nh : List.prod l = x\n⊢ ∀ (y : R), y ∈ l → y ∈ s ∨ y = -1",
"tactic": "tauto"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx : R\nh : x ∈ closure s\n⊢ (∀ (t : List R), t ∈ [] → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1) ∧ List.sum (List.map List.prod []) = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝² : R\nh : x✝² ∈ closure s\nx y : R\nx✝¹ :\n (fun x => ∃ L, (∀ (t : List R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1) ∧ List.sum (List.map List.prod L) = x) x\nx✝ : (fun x => ∃ L, (∀ (t : List R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1) ∧ List.sum (List.map List.prod L) = x) y\nl : List (List R)\nhl1 : ∀ (t : List R), t ∈ l → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1\nhl2 : List.sum (List.map List.prod l) = x\nm : List (List R)\nhm1 : ∀ (t : List R), t ∈ m → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1\nhm2 : List.sum (List.map List.prod m) = y\n⊢ List.sum (List.map List.prod (l ++ m)) = x + y",
"tactic": "simp [hl2, hm2]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝¹ : R\nh : x✝¹ ∈ closure s\nx : R\nx✝ : (fun x => ∃ L, (∀ (t : List R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1) ∧ List.sum (List.map List.prod L) = x) x\nL : List (List R)\nhL : (∀ (t : List R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1) ∧ List.sum (List.map List.prod L) = x\n⊢ List.sum (List.map List.prod (List.map (List.cons (-1)) [])) = -List.sum (List.map List.prod [])",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : Ring T\ns : Set R\nx✝¹ : R\nh : x✝¹ ∈ closure s\nx : R\nx✝ : (fun x => ∃ L, (∀ (t : List R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1) ∧ List.sum (List.map List.prod L) = x) x\nL : List (List R)\nhL : (∀ (t : List R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s ∨ y = -1) ∧ List.sum (List.map List.prod L) = x\n⊢ ∀ (head : List R) (tail : List (List R)),\n List.sum (List.map List.prod (List.map (List.cons (-1)) tail)) = -List.sum (List.map List.prod tail) →\n List.sum (List.map List.prod (List.map (List.cons (-1)) (head :: tail))) =\n -List.sum (List.map List.prod (head :: tail))",
"tactic": "simp (config := { contextual := true }) [List.map_cons, add_comm]"
}
] |
[
1020,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1004,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
norm_smul_of_nonneg
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.12618\nβ : Type u_1\nγ : Type ?u.12624\nι : Type ?u.12627\ninst✝² : NormedField α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nt : ℝ\nht : 0 ≤ t\nx : β\n⊢ ‖t • x‖ = t * ‖x‖",
"tactic": "rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg ht]"
}
] |
[
84,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isClosedMap_snd_of_compactSpace
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs : IsClosed s\n⊢ ∀ (a : Y), a ∈ (Prod.snd '' s)ᶜ → (Prod.snd '' s)ᶜ ∈ 𝓝 a",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs : IsClosed s\n⊢ IsClosed (Prod.snd '' s)",
"tactic": "rw [← isOpen_compl_iff, isOpen_iff_mem_nhds]"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\n⊢ (Prod.snd '' s)ᶜ ∈ 𝓝 y",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs : IsClosed s\n⊢ ∀ (a : Y), a ∈ (Prod.snd '' s)ᶜ → (Prod.snd '' s)ᶜ ∈ 𝓝 a",
"tactic": "intro y hy"
},
{
"state_after": "case this\nα : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\n⊢ univ ×ˢ {y} ⊆ sᶜ\n\nα : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\nthis : univ ×ˢ {y} ⊆ sᶜ\n⊢ (Prod.snd '' s)ᶜ ∈ 𝓝 y",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\n⊢ (Prod.snd '' s)ᶜ ∈ 𝓝 y",
"tactic": "have : univ ×ˢ {y} ⊆ sᶜ"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs✝ : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\nthis : univ ×ˢ {y} ⊆ sᶜ\nU : Set X\nV : Set Y\nhVo : IsOpen V\nhU : univ ⊆ U\nhV : {y} ⊆ V\nhs : U ×ˢ V ⊆ sᶜ\n⊢ (Prod.snd '' s)ᶜ ∈ 𝓝 y",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\nthis : univ ×ˢ {y} ⊆ sᶜ\n⊢ (Prod.snd '' s)ᶜ ∈ 𝓝 y",
"tactic": "rcases generalized_tube_lemma isCompact_univ isCompact_singleton hs.isOpen_compl this\n with ⟨U, V, -, hVo, hU, hV, hs⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs✝ : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\nthis : univ ×ˢ {y} ⊆ sᶜ\nU : Set X\nV : Set Y\nhVo : IsOpen V\nhU : univ ⊆ U\nhV : {y} ⊆ V\nhs : U ×ˢ V ⊆ sᶜ\n⊢ V ⊆ (Prod.snd '' s)ᶜ",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs✝ : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\nthis : univ ×ˢ {y} ⊆ sᶜ\nU : Set X\nV : Set Y\nhVo : IsOpen V\nhU : univ ⊆ U\nhV : {y} ⊆ V\nhs : U ×ˢ V ⊆ sᶜ\n⊢ (Prod.snd '' s)ᶜ ∈ 𝓝 y",
"tactic": "refine mem_nhds_iff.2 ⟨V, ?_, hVo, hV rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs✝ : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\nthis : univ ×ˢ {y} ⊆ sᶜ\nU : Set X\nV : Set Y\nhVo : IsOpen V\nhU : univ ⊆ U\nhV : {y} ⊆ V\nhs : U ×ˢ V ⊆ sᶜ\nz : X × Y\nhzs : z ∈ s\nhzV : z.snd ∈ V\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs✝ : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\nthis : univ ×ˢ {y} ⊆ sᶜ\nU : Set X\nV : Set Y\nhVo : IsOpen V\nhU : univ ⊆ U\nhV : {y} ⊆ V\nhs : U ×ˢ V ⊆ sᶜ\n⊢ V ⊆ (Prod.snd '' s)ᶜ",
"tactic": "rintro _ hzV ⟨z, hzs, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs✝ : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\nthis : univ ×ˢ {y} ⊆ sᶜ\nU : Set X\nV : Set Y\nhVo : IsOpen V\nhU : univ ⊆ U\nhV : {y} ⊆ V\nhs : U ×ˢ V ⊆ sᶜ\nz : X × Y\nhzs : z ∈ s\nhzV : z.snd ∈ V\n⊢ False",
"tactic": "exact hs ⟨hU trivial, hzV⟩ hzs"
},
{
"state_after": "no goals",
"state_before": "case this\nα : Type u\nβ : Type v\nι : Type ?u.90929\nπ : ι → Type ?u.90934\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ns✝ t : Set α\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nY : Type u_2\ninst✝ : TopologicalSpace Y\ns : Set (X × Y)\nhs : IsClosed s\ny : Y\nhy : y ∈ (Prod.snd '' s)ᶜ\n⊢ univ ×ˢ {y} ⊆ sᶜ",
"tactic": "exact fun (x, y') ⟨_, rfl⟩ hs => hy ⟨(x, y'), hs, rfl⟩"
}
] |
[
867,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
857,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
IsBoundedBilinearMap.isBoundedLinearMap_deriv
|
[] |
[
527,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
LinearEquiv.closedEmbedding_of_injective
|
[
{
"state_after": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nf : E →ₗ[𝕜] F\nhf : LinearMap.ker f = ⊥\ninst✝ : FiniteDimensional 𝕜 E\ng : E ≃ₗ[𝕜] { x // x ∈ LinearMap.range f } := ofInjective f (_ : Function.Injective ↑f)\nsrc✝ : Embedding (Subtype.val ∘ ↑(ContinuousLinearEquiv.toHomeomorph (toContinuousLinearEquiv g))) :=\n Embedding.comp embedding_subtype_val\n (Homeomorph.embedding (ContinuousLinearEquiv.toHomeomorph (toContinuousLinearEquiv g)))\nthis : FiniteDimensional 𝕜 { x // x ∈ LinearMap.range f }\n⊢ IsClosed (range ↑f)",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nf : E →ₗ[𝕜] F\nhf : LinearMap.ker f = ⊥\ninst✝ : FiniteDimensional 𝕜 E\ng : E ≃ₗ[𝕜] { x // x ∈ LinearMap.range f } := ofInjective f (_ : Function.Injective ↑f)\nsrc✝ : Embedding (Subtype.val ∘ ↑(ContinuousLinearEquiv.toHomeomorph (toContinuousLinearEquiv g))) :=\n Embedding.comp embedding_subtype_val\n (Homeomorph.embedding (ContinuousLinearEquiv.toHomeomorph (toContinuousLinearEquiv g)))\n⊢ IsClosed (range ↑f)",
"tactic": "haveI := f.finiteDimensional_range"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\nf : E →ₗ[𝕜] F\nhf : LinearMap.ker f = ⊥\ninst✝ : FiniteDimensional 𝕜 E\ng : E ≃ₗ[𝕜] { x // x ∈ LinearMap.range f } := ofInjective f (_ : Function.Injective ↑f)\nsrc✝ : Embedding (Subtype.val ∘ ↑(ContinuousLinearEquiv.toHomeomorph (toContinuousLinearEquiv g))) :=\n Embedding.comp embedding_subtype_val\n (Homeomorph.embedding (ContinuousLinearEquiv.toHomeomorph (toContinuousLinearEquiv g)))\nthis : FiniteDimensional 𝕜 { x // x ∈ LinearMap.range f }\n⊢ IsClosed (range ↑f)",
"tactic": "simpa [LinearMap.range_coe f] using f.range.closed_of_finiteDimensional"
}
] |
[
524,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
518,
1
] |
Mathlib/MeasureTheory/Function/Egorov.lean
|
MeasureTheory.Egorov.notConvergentSeq_antitone
|
[] |
[
59,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
|
isLittleO_zpow_exp_pos_mul_atTop
|
[
{
"state_after": "no goals",
"state_before": "k : ℤ\nb : ℝ\nhb : 0 < b\n⊢ (fun x => x ^ ↑k) =o[atTop] fun x => exp (b * x)",
"tactic": "simpa only [rpow_int_cast] using isLittleO_rpow_exp_pos_mul_atTop k hb"
}
] |
[
258,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieIdeal.map_comap_eq
|
[
{
"state_after": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ map f (comap f J) ≤ LieHom.idealRange f ⊓ J\n\ncase a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ LieHom.idealRange f ⊓ J ≤ map f (comap f J)",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ map f (comap f J) = LieHom.idealRange f ⊓ J",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ map f (comap f J) ≤ LieHom.idealRange f ∧ map f (comap f J) ≤ J",
"state_before": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ map f (comap f J) ≤ LieHom.idealRange f ⊓ J",
"tactic": "rw [le_inf_iff]"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ map f (comap f J) ≤ LieHom.idealRange f ∧ map f (comap f J) ≤ J",
"tactic": "exact ⟨f.map_le_idealRange _, map_comap_le⟩"
},
{
"state_after": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑R L' (LieHom.idealRange f) = LieHom.range f\n⊢ LieHom.idealRange f ⊓ J ≤ map f (comap f J)",
"state_before": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ LieHom.idealRange f ⊓ J ≤ map f (comap f J)",
"tactic": "rw [f.isIdealMorphism_def] at h"
},
{
"state_after": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑R L' (LieHom.idealRange f) = LieHom.range f\n⊢ ↑(LieHom.range f) ∩ ↑J ⊆ ↑(map f (comap f J))",
"state_before": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑R L' (LieHom.idealRange f) = LieHom.range f\n⊢ LieHom.idealRange f ⊓ J ≤ map f (comap f J)",
"tactic": "rw [← SetLike.coe_subset_coe, LieSubmodule.inf_coe, ← coe_toSubalgebra, h]"
},
{
"state_after": "case a.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑R L' (LieHom.idealRange f) = LieHom.range f\ny : L'\nh₂ : y ∈ ↑J\nx : L\nh₁ : ↑↑f x = y\n⊢ y ∈ ↑(map f (comap f J))",
"state_before": "case a\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑R L' (LieHom.idealRange f) = LieHom.range f\n⊢ ↑(LieHom.range f) ∩ ↑J ⊆ ↑(map f (comap f J))",
"tactic": "rintro y ⟨⟨x, h₁⟩, h₂⟩"
},
{
"state_after": "case a.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑R L' (LieHom.idealRange f) = LieHom.range f\ny : L'\nx : L\nh₂ : ↑↑f x ∈ ↑J\nh₁ : ↑↑f x = y\n⊢ ↑↑f x ∈ ↑(map f (comap f J))",
"state_before": "case a.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑R L' (LieHom.idealRange f) = LieHom.range f\ny : L'\nh₂ : y ∈ ↑J\nx : L\nh₁ : ↑↑f x = y\n⊢ y ∈ ↑(map f (comap f J))",
"tactic": "rw [← h₁] at h₂⊢"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : ↑R L' (LieHom.idealRange f) = LieHom.range f\ny : L'\nx : L\nh₂ : ↑↑f x ∈ ↑J\nh₁ : ↑↑f x = y\n⊢ ↑↑f x ∈ ↑(map f (comap f J))",
"tactic": "exact mem_map h₂"
}
] |
[
1092,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1087,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
Isometry.edist_eq
|
[] |
[
81,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.one_le_blocks'
|
[] |
[
187,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Algebra/Homology/Additive.lean
|
HomologicalComplex.singleMapHomologicalComplex_inv_app_self
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.172926\nV : Type u\ninst✝⁶ : Category V\ninst✝⁵ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\ninst✝⁴ : HasZeroObject V\nW : Type ?u.173295\ninst✝³ : Category W\ninst✝² : Preadditive W\ninst✝¹ : HasZeroObject W\nF : V ⥤ W\ninst✝ : Functor.Additive F\nj : ι\nX : V\n⊢ HomologicalComplex.X ((F ⋙ single W c j).obj X) j =\n HomologicalComplex.X ((single V c j ⋙ Functor.mapHomologicalComplex F c).obj X) j",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_3\nV : Type u\ninst✝⁶ : Category V\ninst✝⁵ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\ninst✝⁴ : HasZeroObject V\nW : Type u_2\ninst✝³ : Category W\ninst✝² : Preadditive W\ninst✝¹ : HasZeroObject W\nF : V ⥤ W\ninst✝ : Functor.Additive F\nj : ι\nX : V\n⊢ Hom.f ((singleMapHomologicalComplex F c j).inv.app X) j =\n eqToHom (_ : (if j = j then F.obj X else 0) = F.obj (if j = j then X else 0))",
"tactic": "simp [singleMapHomologicalComplex]"
}
] |
[
330,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.coeff_zero
|
[] |
[
184,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.isClosed_image_coe_of_bounded_of_weak_closed
|
[] |
[
1629,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1624,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.moveLeft_lf
|
[] |
[
598,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
597,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiffOn_snd
|
[] |
[
812,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
811,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean
|
mem_posTangentConeAt_of_segment_subset
|
[
{
"state_after": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\n⊢ y - x ∈ posTangentConeAt s x",
"state_before": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\n⊢ y - x ∈ posTangentConeAt s x",
"tactic": "let c := fun n : ℕ => (2 : ℝ) ^ n"
},
{
"state_after": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\n⊢ y - x ∈ posTangentConeAt s x",
"state_before": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\n⊢ y - x ∈ posTangentConeAt s x",
"tactic": "let d := fun n : ℕ => (c n)⁻¹ • (y - x)"
},
{
"state_after": "case refine'_1\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\nn : ℕ\n⊢ x + d n ∈ segment ℝ x y\n\ncase refine'_2\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\n⊢ Tendsto (fun n => c n • d n) atTop (𝓝 (y - x))",
"state_before": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\n⊢ y - x ∈ posTangentConeAt s x",
"tactic": "refine' ⟨c, d, Filter.univ_mem' fun n => h _, tendsto_pow_atTop_atTop_of_one_lt one_lt_two, _⟩"
},
{
"state_after": "case refine'_1\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\nn : ℕ\n⊢ x + d n ∈ segment ℝ x y\n\ncase refine'_2\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\n⊢ Tendsto (fun n => c n • d n) atTop (𝓝 (y - x))",
"state_before": "case refine'_1\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\nn : ℕ\n⊢ x + d n ∈ segment ℝ x y\n\ncase refine'_2\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\n⊢ Tendsto (fun n => c n • d n) atTop (𝓝 (y - x))",
"tactic": "show x + d n ∈ segment ℝ x y"
},
{
"state_after": "case refine'_2\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\n⊢ Tendsto (fun n => c n • d n) atTop (𝓝 (y - x))",
"state_before": "case refine'_2\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\n⊢ Tendsto (fun n => c n • d n) atTop (𝓝 (y - x))",
"tactic": "show Tendsto (fun n => c n • d n) atTop (𝓝 (y - x))"
},
{
"state_after": "case refine'_1\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\nn : ℕ\n⊢ x + d n ∈ (fun θ => x + θ • (y - x)) '' Icc 0 1",
"state_before": "case refine'_1\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\nn : ℕ\n⊢ x + d n ∈ segment ℝ x y",
"tactic": "rw [segment_eq_image']"
},
{
"state_after": "case refine'_1.refine'_1\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\nn : ℕ\n⊢ 0 ≤ (c n)⁻¹\n\ncase refine'_1.refine'_2\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\nn : ℕ\n⊢ (c n)⁻¹ ≤ 1",
"state_before": "case refine'_1\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\nn : ℕ\n⊢ x + d n ∈ (fun θ => x + θ • (y - x)) '' Icc 0 1",
"tactic": "refine' ⟨(c n)⁻¹, ⟨_, _⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.refine'_1\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\nn : ℕ\n⊢ 0 ≤ (c n)⁻¹\n\ncase refine'_1.refine'_2\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\nn : ℕ\n⊢ (c n)⁻¹ ≤ 1",
"tactic": "exacts [inv_nonneg.2 (pow_nonneg zero_le_two _), inv_le_one (one_le_pow_of_one_le one_le_two _)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\na : E\nf' : E →L[ℝ] ℝ\ns : Set E\nx y : E\nh : segment ℝ x y ⊆ s\nc : ℕ → ℝ := fun n => 2 ^ n\nd : ℕ → E := fun n => (c n)⁻¹ • (y - x)\n⊢ Tendsto (fun n => c n • d n) atTop (𝓝 (y - x))",
"tactic": "exact tendsto_const_nhds.congr fun n ↦ (smul_inv_smul₀ (pow_ne_zero _ two_ne_zero) _).symm"
}
] |
[
102,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMax.inf
|
[] |
[
482,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
480,
8
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
normalizerCondition_iff_only_full_group_self_normalizing
|
[
{
"state_after": "case h\nG : Type u_1\nG' : Type ?u.405590\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.405599\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type ?u.405617\ninst✝ : Group N\n⊢ ∀ (a : Subgroup G), a < ⊤ → a < normalizer a ↔ normalizer a = a → a = ⊤",
"state_before": "G : Type u_1\nG' : Type ?u.405590\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.405599\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type ?u.405617\ninst✝ : Group N\n⊢ NormalizerCondition G ↔ ∀ (H : Subgroup G), normalizer H = H → H = ⊤",
"tactic": "apply forall_congr'"
},
{
"state_after": "case h\nG : Type u_1\nG' : Type ?u.405590\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.405599\ninst✝¹ : AddGroup A\nH✝ K : Subgroup G\nN : Type ?u.405617\ninst✝ : Group N\nH : Subgroup G\n⊢ H < ⊤ → H < normalizer H ↔ normalizer H = H → H = ⊤",
"state_before": "case h\nG : Type u_1\nG' : Type ?u.405590\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.405599\ninst✝¹ : AddGroup A\nH K : Subgroup G\nN : Type ?u.405617\ninst✝ : Group N\n⊢ ∀ (a : Subgroup G), a < ⊤ → a < normalizer a ↔ normalizer a = a → a = ⊤",
"tactic": "intro H"
},
{
"state_after": "case h\nG : Type u_1\nG' : Type ?u.405590\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.405599\ninst✝¹ : AddGroup A\nH✝ K : Subgroup G\nN : Type ?u.405617\ninst✝ : Group N\nH : Subgroup G\n⊢ ¬H = ⊤ → ¬H = normalizer H ↔ normalizer H = H → H = ⊤",
"state_before": "case h\nG : Type u_1\nG' : Type ?u.405590\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.405599\ninst✝¹ : AddGroup A\nH✝ K : Subgroup G\nN : Type ?u.405617\ninst✝ : Group N\nH : Subgroup G\n⊢ H < ⊤ → H < normalizer H ↔ normalizer H = H → H = ⊤",
"tactic": "simp only [lt_iff_le_and_ne, le_normalizer, true_and_iff, le_top, Ne.def]"
},
{
"state_after": "no goals",
"state_before": "case h\nG : Type u_1\nG' : Type ?u.405590\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.405599\ninst✝¹ : AddGroup A\nH✝ K : Subgroup G\nN : Type ?u.405617\ninst✝ : Group N\nH : Subgroup G\n⊢ ¬H = ⊤ → ¬H = normalizer H ↔ normalizer H = H → H = ⊤",
"tactic": "tauto"
}
] |
[
2251,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2247,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Odd.not_two_dvd_nat
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\nh : Odd n\n⊢ ¬2 ∣ n",
"tactic": "rwa [← even_iff_two_dvd, ← odd_iff_not_even]"
}
] |
[
68,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/CategoryTheory/Preadditive/Generator.lean
|
CategoryTheory.isCoseparator_iff_faithful_preadditiveYonedaObj
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\nG : C\n⊢ Faithful (preadditiveYonedaObj G ⋙ forget₂ (ModuleCat (End G)) AddCommGroupCat) ↔ Faithful (preadditiveYonedaObj G)",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\nG : C\n⊢ IsCoseparator G ↔ Faithful (preadditiveYonedaObj G)",
"tactic": "rw [isCoseparator_iff_faithful_preadditiveYoneda, preadditiveYoneda_obj]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Preadditive C\nG : C\n⊢ Faithful (preadditiveYonedaObj G ⋙ forget₂ (ModuleCat (End G)) AddCommGroupCat) ↔ Faithful (preadditiveYonedaObj G)",
"tactic": "exact ⟨fun h => Faithful.of_comp _ (forget₂ _ AddCommGroupCat.{v}), fun h => Faithful.comp _ _⟩"
}
] |
[
80,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventually_iff
|
[] |
[
1070,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1069,
1
] |
Mathlib/Algebra/Homology/Exact.lean
|
CategoryTheory.comp_eq_zero_of_exact
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasEqualizers V\ninst✝ : HasCokernels V\nh : Exact f g\nX Y : V\nι : X ⟶ B\nhι : ι ≫ g = 0\nπ : B ⟶ Y\nhπ : f ≫ π = 0\n⊢ ι ≫ π = 0",
"tactic": "rw [← kernel.lift_ι _ _ hι, ← cokernel.π_desc _ _ hπ, Category.assoc,\n kernel_comp_cokernel_assoc _ _ h, zero_comp, comp_zero]"
}
] |
[
286,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.insert_diff_singleton
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na✝ b : α\ns✝ s₁ s₂ t t₁ t₂ u : Set α\na : α\ns : Set α\n⊢ insert a (s \\ {a}) = insert a s",
"tactic": "simp [insert_eq, union_diff_self, -union_singleton, -singleton_union]"
}
] |
[
2055,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2054,
1
] |
Mathlib/Data/List/Permutation.lean
|
List.map_permutationsAux
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → β\n⊢ ∀ (t : α) (ts is : List α),\n map (map f) (permutationsAux ts (t :: is)) = permutationsAux (map f ts) (map f (t :: is)) →\n map (map f) (permutationsAux is []) = permutationsAux (map f is) (map f []) →\n map (map f) (permutationsAux (t :: ts) is) = permutationsAux (map f (t :: ts)) (map f is)",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β\n⊢ ∀ (ts is : List α), map (map f) (permutationsAux ts is) = permutationsAux (map f ts) (map f is)",
"tactic": "refine' permutationsAux.rec (by simp) _"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → β\nt : α\nts is : List α\nIH1 : map (map f) (permutationsAux ts (t :: is)) = permutationsAux (map f ts) (map f (t :: is))\nIH2 : map (map f) (permutationsAux is []) = permutationsAux (map f is) (map f [])\n⊢ map (map f) (permutationsAux (t :: ts) is) = permutationsAux (map f (t :: ts)) (map f is)",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β\n⊢ ∀ (t : α) (ts is : List α),\n map (map f) (permutationsAux ts (t :: is)) = permutationsAux (map f ts) (map f (t :: is)) →\n map (map f) (permutationsAux is []) = permutationsAux (map f is) (map f []) →\n map (map f) (permutationsAux (t :: ts) is) = permutationsAux (map f (t :: ts)) (map f is)",
"tactic": "introv IH1 IH2"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nf : α → β\nt : α\nts is : List α\nIH1 : map (map f) (permutationsAux ts (t :: is)) = permutationsAux (map f ts) (map f (t :: is))\nIH2 : map (map f) (permutationsAux is []) = permutationsAux (map f is) []\n⊢ map (map f) (permutationsAux (t :: ts) is) = permutationsAux (map f (t :: ts)) (map f is)",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β\nt : α\nts is : List α\nIH1 : map (map f) (permutationsAux ts (t :: is)) = permutationsAux (map f ts) (map f (t :: is))\nIH2 : map (map f) (permutationsAux is []) = permutationsAux (map f is) (map f [])\n⊢ map (map f) (permutationsAux (t :: ts) is) = permutationsAux (map f (t :: ts)) (map f is)",
"tactic": "rw [map] at IH2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β\nt : α\nts is : List α\nIH1 : map (map f) (permutationsAux ts (t :: is)) = permutationsAux (map f ts) (map f (t :: is))\nIH2 : map (map f) (permutationsAux is []) = permutationsAux (map f is) []\n⊢ map (map f) (permutationsAux (t :: ts) is) = permutationsAux (map f (t :: ts)) (map f is)",
"tactic": "simp only [foldr_permutationsAux2, map_append, map, map_map_permutationsAux2, permutations,\n bind_map, IH1, append_assoc, permutationsAux_cons, cons_bind, ← IH2, map_bind]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nf : α → β\n⊢ ∀ (is : List α), map (map f) (permutationsAux [] is) = permutationsAux (map f []) (map f is)",
"tactic": "simp"
}
] |
[
239,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/Topology/Order.lean
|
TopologicalSpace.tendsto_nhds_generateFrom
|
[
{
"state_after": "α : Type u\nβ : Type u_1\nm : α → β\nf : Filter α\ng : Set (Set β)\nb : β\nh : ∀ (s : Set β), s ∈ g → b ∈ s → m ⁻¹' s ∈ f\n⊢ Tendsto m f (⨅ (s : Set β) (_ : s ∈ {s | b ∈ s ∧ s ∈ g}), 𝓟 s)",
"state_before": "α : Type u\nβ : Type u_1\nm : α → β\nf : Filter α\ng : Set (Set β)\nb : β\nh : ∀ (s : Set β), s ∈ g → b ∈ s → m ⁻¹' s ∈ f\n⊢ Tendsto m f (𝓝 b)",
"tactic": "rw [nhds_generateFrom]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type u_1\nm : α → β\nf : Filter α\ng : Set (Set β)\nb : β\nh : ∀ (s : Set β), s ∈ g → b ∈ s → m ⁻¹' s ∈ f\n⊢ Tendsto m f (⨅ (s : Set β) (_ : s ∈ {s | b ∈ s ∧ s ∈ g}), 𝓟 s)",
"tactic": "exact tendsto_iInf.2 fun s => tendsto_iInf.2 fun ⟨hbs, hsg⟩ => tendsto_principal.2 <| h s hsg hbs"
}
] |
[
99,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/MeasureTheory/Group/Integration.lean
|
MeasureTheory.lintegral_mul_right_eq_self
|
[
{
"state_after": "case h.e'_3\n𝕜 : Type ?u.26943\nM : Type ?u.26946\nα : Type ?u.26949\nG : Type u_1\nE : Type ?u.26955\nF : Type ?u.26958\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\nμ : Measure G\nf✝ : G → E\ng✝ : G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : IsMulRightInvariant μ\nf : G → ℝ≥0∞\ng : G\n⊢ (∫⁻ (x : G), f x ∂μ) = ∫⁻ (a : G), f a ∂map (↑(MeasurableEquiv.mulRight g)) μ",
"state_before": "𝕜 : Type ?u.26943\nM : Type ?u.26946\nα : Type ?u.26949\nG : Type u_1\nE : Type ?u.26955\nF : Type ?u.26958\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\nμ : Measure G\nf✝ : G → E\ng✝ : G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : IsMulRightInvariant μ\nf : G → ℝ≥0∞\ng : G\n⊢ (∫⁻ (x : G), f (x * g) ∂μ) = ∫⁻ (x : G), f x ∂μ",
"tactic": "convert (lintegral_map_equiv f <| MeasurableEquiv.mulRight g).symm using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\n𝕜 : Type ?u.26943\nM : Type ?u.26946\nα : Type ?u.26949\nG : Type u_1\nE : Type ?u.26955\nF : Type ?u.26958\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\nμ : Measure G\nf✝ : G → E\ng✝ : G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : IsMulRightInvariant μ\nf : G → ℝ≥0∞\ng : G\n⊢ (∫⁻ (x : G), f x ∂μ) = ∫⁻ (a : G), f a ∂map (↑(MeasurableEquiv.mulRight g)) μ",
"tactic": "simp [map_mul_right_eq_self μ g]"
}
] |
[
80,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
LinearEquiv.funUnique_apply
|
[] |
[
487,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.Red.Step.length
|
[
{
"state_after": "α : Type u\nL L₁ L₂ L₃ L₄ L1 L2 : List (α × Bool)\nx : α\nb : Bool\n⊢ List.length L1 + List.length L2 + 2 = List.length L1 + List.length ((x, b) :: (x, !b) :: L2)",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ L1 L2 : List (α × Bool)\nx : α\nb : Bool\n⊢ List.length (L1 ++ L2) + 2 = List.length (L1 ++ (x, b) :: (x, !b) :: L2)",
"tactic": "rw [List.length_append, List.length_append]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ L1 L2 : List (α × Bool)\nx : α\nb : Bool\n⊢ List.length L1 + List.length L2 + 2 = List.length L1 + List.length ((x, b) :: (x, !b) :: L2)",
"tactic": "rfl"
}
] |
[
113,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
|
MeasureTheory.ProbabilityMeasure.toWeakDualBCNN_apply
|
[] |
[
245,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
GroupNorm.coe_sup
|
[] |
[
837,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
836,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.gc_map_comap
|
[] |
[
741,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
741,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.eq
|
[] |
[
752,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
752,
11
] |
Std/Data/Nat/Lemmas.lean
|
Nat.log2_lt
|
[
{
"state_after": "no goals",
"state_before": "n k : Nat\nh : n ≠ 0\n⊢ log2 n < k ↔ n < 2 ^ k",
"tactic": "rw [← Nat.not_le, ← Nat.not_le, le_log2 h]"
}
] |
[
823,
45
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
822,
1
] |
Mathlib/Data/Nat/Factors.lean
|
Nat.factors_sorted
|
[] |
[
116,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean
|
Complex.isOpenMap_exp
|
[] |
[
28,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
27,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.pred_lt_pred_iff
|
[
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\na b : Fin (Nat.succ n)\nha : a ≠ 0\nhb : b ≠ 0\n⊢ pred a ha < pred b hb ↔ a < b",
"tactic": "rw [← succ_lt_succ_iff, succ_pred, succ_pred]"
}
] |
[
1548,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1547,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
|
LiouvilleWith.irrational
|
[
{
"state_after": "case intro\np q y : ℝ\nr✝ : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nr : ℚ\nh : LiouvilleWith p ↑r\n⊢ False",
"state_before": "p q x y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\nh : LiouvilleWith p x\nhp : 1 < p\n⊢ Irrational x",
"tactic": "rintro ⟨r, rfl⟩"
},
{
"state_after": "case intro.inl\np q y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nh : LiouvilleWith p ↑0\n⊢ False\n\ncase intro.inr\np q y : ℝ\nr✝ : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nr : ℚ\nh : LiouvilleWith p ↑r\nh0 : r ≠ 0\n⊢ False",
"state_before": "case intro\np q y : ℝ\nr✝ : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nr : ℚ\nh : LiouvilleWith p ↑r\n⊢ False",
"tactic": "rcases eq_or_ne r 0 with (rfl | h0)"
},
{
"state_after": "case intro.inl\np q y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nh : LiouvilleWith p ↑0\n⊢ ↑0 = ↑0",
"state_before": "case intro.inl\np q y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nh : LiouvilleWith p ↑0\n⊢ False",
"tactic": "refine h.ne_cast_int hp 0 ?_"
},
{
"state_after": "no goals",
"state_before": "case intro.inl\np q y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nh : LiouvilleWith p ↑0\n⊢ ↑0 = ↑0",
"tactic": "rw [Rat.cast_zero, Int.cast_zero]"
},
{
"state_after": "case intro.inr\np q y : ℝ\nr✝ : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nr : ℚ\nh : LiouvilleWith p ↑r\nh0 : r ≠ 0\n⊢ ↑r * ↑r⁻¹ = ↑1",
"state_before": "case intro.inr\np q y : ℝ\nr✝ : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nr : ℚ\nh : LiouvilleWith p ↑r\nh0 : r ≠ 0\n⊢ False",
"tactic": "refine (h.mul_rat (inv_ne_zero h0)).ne_cast_int hp 1 ?_"
},
{
"state_after": "case intro.inr\np q y : ℝ\nr✝ : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nr : ℚ\nh : LiouvilleWith p ↑r\nh0 : r ≠ 0\n⊢ 1 = ↑1\n\ncase intro.inr\np q y : ℝ\nr✝ : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nr : ℚ\nh : LiouvilleWith p ↑r\nh0 : r ≠ 0\n⊢ ↑r ≠ 0",
"state_before": "case intro.inr\np q y : ℝ\nr✝ : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nr : ℚ\nh : LiouvilleWith p ↑r\nh0 : r ≠ 0\n⊢ ↑r * ↑r⁻¹ = ↑1",
"tactic": "rw [Rat.cast_inv, mul_inv_cancel]"
},
{
"state_after": "no goals",
"state_before": "case intro.inr\np q y : ℝ\nr✝ : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nr : ℚ\nh : LiouvilleWith p ↑r\nh0 : r ≠ 0\n⊢ 1 = ↑1\n\ncase intro.inr\np q y : ℝ\nr✝ : ℚ\nm : ℤ\nn : ℕ\nhp : 1 < p\nr : ℚ\nh : LiouvilleWith p ↑r\nh0 : r ≠ 0\n⊢ ↑r ≠ 0",
"tactic": "exacts [Int.cast_one.symm, Rat.cast_ne_zero.mpr h0]"
}
] |
[
334,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
11
] |
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