file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Analysis/Normed/Group/Basic.lean
|
pi_norm_const'
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.1275542\n𝕜 : Type ?u.1275545\nα : Type ?u.1275548\nι : Type u_1\nκ : Type ?u.1275554\nE : Type u_2\nF : Type ?u.1275560\nG : Type ?u.1275563\nπ : ι → Type ?u.1275568\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → SeminormedGroup (π i)\ninst✝¹ : SeminormedGroup E\nf x : (i : ι) → π i\nr : ℝ\ninst✝ : Nonempty ι\na : E\n⊢ ‖fun _i => a‖ = ‖a‖",
"tactic": "simpa only [← dist_one_right] using dist_pi_const a 1"
}
] |
[
2535,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2534,
1
] |
Mathlib/Data/Nat/Count.lean
|
Nat.count_add'
|
[
{
"state_after": "p : ℕ → Prop\ninst✝ : DecidablePred p\na b : ℕ\n⊢ count (fun k => p (b + k)) a + count p b = count (fun k => p (k + b)) a + count p b",
"state_before": "p : ℕ → Prop\ninst✝ : DecidablePred p\na b : ℕ\n⊢ count p (a + b) = count (fun k => p (k + b)) a + count p b",
"tactic": "rw [add_comm, count_add, add_comm]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ → Prop\ninst✝ : DecidablePred p\na b : ℕ\n⊢ count (fun k => p (b + k)) a + count p b = count (fun k => p (k + b)) a + count p b",
"tactic": "simp_rw [add_comm b]"
}
] |
[
91,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Data/Set/Countable.lean
|
Set.countable_univ_pi
|
[] |
[
270,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.powers_one
|
[] |
[
455,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
hasFDerivAtFilter_iff_hasDerivAtFilter
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf'✝ f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nf' : 𝕜 →L[𝕜] F\n⊢ HasFDerivAtFilter f f' x L ↔ HasDerivAtFilter f (↑f' 1) x L",
"tactic": "simp [HasDerivAtFilter]"
}
] |
[
163,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.IsInvertedBy.leftOp
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nW : MorphismProperty C\nL : C ⥤ Dᵒᵖ\nh : IsInvertedBy W L\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : MorphismProperty.op W f\nthis : IsIso (L.map f.unop)\n⊢ IsIso (L.leftOp.map f)",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nW : MorphismProperty C\nL : C ⥤ Dᵒᵖ\nh : IsInvertedBy W L\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : MorphismProperty.op W f\n⊢ IsIso (L.leftOp.map f)",
"tactic": "haveI := h f.unop hf"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nW : MorphismProperty C\nL : C ⥤ Dᵒᵖ\nh : IsInvertedBy W L\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : MorphismProperty.op W f\nthis : IsIso (L.map f.unop)\n⊢ IsIso (L.map f.unop).unop",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nW : MorphismProperty C\nL : C ⥤ Dᵒᵖ\nh : IsInvertedBy W L\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : MorphismProperty.op W f\nthis : IsIso (L.map f.unop)\n⊢ IsIso (L.leftOp.map f)",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u_2\ninst✝ : Category D\nW : MorphismProperty C\nL : C ⥤ Dᵒᵖ\nh : IsInvertedBy W L\nX Y : Cᵒᵖ\nf : X ⟶ Y\nhf : MorphismProperty.op W f\nthis : IsIso (L.map f.unop)\n⊢ IsIso (L.map f.unop).unop",
"tactic": "infer_instance"
}
] |
[
328,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.add_ne_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.82044\nβ : Type ?u.82047\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ a + b ≠ ⊤ ↔ a ≠ ⊤ ∧ b ≠ ⊤",
"tactic": "simpa only [lt_top_iff_ne_top] using add_lt_top"
}
] |
[
546,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.repr_range
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.444864\nR : Type u_1\nK : Type ?u.444870\nM : Type u_2\nM' : Type ?u.444876\nM'' : Type ?u.444879\nV : Type u\nV' : Type ?u.444884\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\n⊢ LinearMap.range (repr hv) = ⊤",
"tactic": "rw [LinearIndependent.repr, LinearEquiv.range]"
}
] |
[
795,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
794,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.zero_of_eval_zero
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q : R[X]\ninst✝ : Infinite R\np : R[X]\nh : ∀ (x : R), eval x p = 0\nhp : ¬p = 0\n⊢ False",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q : R[X]\ninst✝ : Infinite R\np : R[X]\nh : ∀ (x : R), eval x p = 0\n⊢ p = 0",
"tactic": "classical by_contra hp;"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q : R[X]\ninst✝ : Infinite R\np : R[X]\nh : ∀ (x : R), eval x p = 0\nhp : ¬p = 0\n⊢ Fintype R",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q : R[X]\ninst✝ : Infinite R\np : R[X]\nh : ∀ (x : R), eval x p = 0\nhp : ¬p = 0\n⊢ False",
"tactic": "refine @Fintype.false R _ ?_"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q : R[X]\ninst✝ : Infinite R\np : R[X]\nh : ∀ (x : R), eval x p = 0\nhp : ¬p = 0\n⊢ Fintype R",
"tactic": "exact ⟨p.roots.toFinset, fun x => Multiset.mem_toFinset.mpr ((mem_roots hp).mpr (h _))⟩"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q : R[X]\ninst✝ : Infinite R\np : R[X]\nh : ∀ (x : R), eval x p = 0\nhp : ¬p = 0\n⊢ False",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q : R[X]\ninst✝ : Infinite R\np : R[X]\nh : ∀ (x : R), eval x p = 0\n⊢ p = 0",
"tactic": "by_contra hp"
}
] |
[
872,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
869,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasurableSet.map_coe_volume
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1818394\nγ : Type ?u.1818397\nδ : Type ?u.1818400\nι : Type ?u.1818403\nR : Type ?u.1818406\nR' : Type ?u.1818409\ninst✝ : MeasureSpace α\ns✝ t s : Set α\nhs : MeasurableSet s\n⊢ Measure.map Subtype.val volume = Measure.restrict volume s",
"tactic": "rw [volume_set_coe_def, (MeasurableEmbedding.subtype_coe hs).map_comap volume, Subtype.range_coe]"
}
] |
[
4261,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4259,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_pos
|
[] |
[
177,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.biInf_mem
|
[] |
[
819,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
814,
11
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullbackRightPullbackFstIso_inv_snd_fst
|
[
{
"state_after": "C : Type u\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nf' : W ⟶ X\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback f' pullback.fst\ninst✝ : HasPullback (f' ≫ f) g\n⊢ (pullbackRightPullbackFstIso f g f').inv ≫ pullback.fst ≫ f' = pullback.fst ≫ f'",
"state_before": "C : Type u\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nf' : W ⟶ X\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback f' pullback.fst\ninst✝ : HasPullback (f' ≫ f) g\n⊢ (pullbackRightPullbackFstIso f g f').inv ≫ pullback.snd ≫ pullback.fst = pullback.fst ≫ f'",
"tactic": "rw [← pullback.condition]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\nf' : W ⟶ X\ninst✝² : HasPullback f g\ninst✝¹ : HasPullback f' pullback.fst\ninst✝ : HasPullback (f' ≫ f) g\n⊢ (pullbackRightPullbackFstIso f g f').inv ≫ pullback.fst ≫ f' = pullback.fst ≫ f'",
"tactic": "exact pullbackRightPullbackFstIso_inv_fst_assoc _ _ _ _"
}
] |
[
2191,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2188,
1
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.IsTopologicalBasis.nhds_hasBasis
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nt : TopologicalSpace α\nb : Set (Set α)\nhb : IsTopologicalBasis b\na : α\ns : Set α\n⊢ (∃ t, t ∈ b ∧ a ∈ t ∧ t ⊆ s) ↔ ∃ i, (i ∈ b ∧ a ∈ i) ∧ i ⊆ s",
"tactic": "simp only [exists_prop, and_assoc]"
}
] |
[
165,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.preimage_cast_Icc
|
[
{
"state_after": "case h\nF : Type ?u.68729\nι : Type ?u.68732\nα : Type ?u.68735\nβ : Type ?u.68738\nK : Type u_1\ninst✝ : LinearOrderedField K\na b x : ℚ\n⊢ x ∈ Rat.cast ⁻¹' Icc ↑a ↑b ↔ x ∈ Icc a b",
"state_before": "F : Type ?u.68729\nι : Type ?u.68732\nα : Type ?u.68735\nβ : Type ?u.68738\nK : Type u_1\ninst✝ : LinearOrderedField K\na b : ℚ\n⊢ Rat.cast ⁻¹' Icc ↑a ↑b = Icc a b",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type ?u.68729\nι : Type ?u.68732\nα : Type ?u.68735\nβ : Type ?u.68738\nK : Type u_1\ninst✝ : LinearOrderedField K\na b x : ℚ\n⊢ x ∈ Rat.cast ⁻¹' Icc ↑a ↑b ↔ x ∈ Icc a b",
"tactic": "simp"
}
] |
[
371,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Order/CompleteLattice.lean
|
unary_relation_sInf_iff
|
[
{
"state_after": "α✝ : Type ?u.198791\nβ : Type ?u.198794\nβ₂ : Type ?u.198797\nγ : Type ?u.198800\nι : Sort ?u.198803\nι' : Sort ?u.198806\nκ : ι → Sort ?u.198811\nκ' : ι' → Sort ?u.198816\nα : Type u_1\ns : Set (α → Prop)\na : α\n⊢ (⨅ (f : ↑s), ↑f a) ↔ ∀ (r : α → Prop), r ∈ s → r a",
"state_before": "α✝ : Type ?u.198791\nβ : Type ?u.198794\nβ₂ : Type ?u.198797\nγ : Type ?u.198800\nι : Sort ?u.198803\nι' : Sort ?u.198806\nκ : ι → Sort ?u.198811\nκ' : ι' → Sort ?u.198816\nα : Type u_1\ns : Set (α → Prop)\na : α\n⊢ sInf s a ↔ ∀ (r : α → Prop), r ∈ s → r a",
"tactic": "rw [sInf_apply]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.198791\nβ : Type ?u.198794\nβ₂ : Type ?u.198797\nγ : Type ?u.198800\nι : Sort ?u.198803\nι' : Sort ?u.198806\nκ : ι → Sort ?u.198811\nκ' : ι' → Sort ?u.198816\nα : Type u_1\ns : Set (α → Prop)\na : α\n⊢ (⨅ (f : ↑s), ↑f a) ↔ ∀ (r : α → Prop), r ∈ s → r a",
"tactic": "simp [← eq_iff_iff]"
}
] |
[
1790,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1787,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
bddBelow_def
|
[] |
[
99,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.mem_powersetAux
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\ns : Multiset α\n⊢ ∀ (a : List α), Quotient.mk (isSetoid α) a ∈ powersetAux l ↔ Quotient.mk (isSetoid α) a ≤ ↑l",
"tactic": "simp [powersetAux_eq_map_coe, Subperm, and_comm]"
}
] |
[
40,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Algebra/Symmetrized.lean
|
SymAlg.unsym_bijective
|
[] |
[
99,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Set/Intervals/UnorderedInterval.lean
|
Set.uIcc_prod_uIcc
|
[] |
[
149,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
fderiv_id'
|
[] |
[
1038,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1037,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.iff_le_add_mul
|
[] |
[
348,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
11
] |
Mathlib/Data/Dfinsupp/NeLocus.lean
|
Dfinsupp.coe_neLocus
|
[] |
[
55,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean
|
Real.Angle.expMapCircle_add
|
[
{
"state_after": "case h\nθ₂ : Angle\nx✝ : ℝ\n⊢ expMapCircle (↑x✝ + θ₂) = expMapCircle ↑x✝ * expMapCircle θ₂",
"state_before": "θ₁ θ₂ : Angle\n⊢ expMapCircle (θ₁ + θ₂) = expMapCircle θ₁ * expMapCircle θ₂",
"tactic": "induction θ₁ using Real.Angle.induction_on"
},
{
"state_after": "case h.h\nx✝¹ x✝ : ℝ\n⊢ expMapCircle (↑x✝¹ + ↑x✝) = expMapCircle ↑x✝¹ * expMapCircle ↑x✝",
"state_before": "case h\nθ₂ : Angle\nx✝ : ℝ\n⊢ expMapCircle (↑x✝ + θ₂) = expMapCircle ↑x✝ * expMapCircle θ₂",
"tactic": "induction θ₂ using Real.Angle.induction_on"
},
{
"state_after": "no goals",
"state_before": "case h.h\nx✝¹ x✝ : ℝ\n⊢ expMapCircle (↑x✝¹ + ↑x✝) = expMapCircle ↑x✝¹ * expMapCircle ↑x✝",
"tactic": "exact _root_.expMapCircle_add _ _"
}
] |
[
146,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Topology/Order/Basic.lean
|
isClosed_Iic
|
[] |
[
139,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.toIsometryEquiv_inj
|
[] |
[
644,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/Algebra/Order/Archimedean.lean
|
exists_int_lt
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nn : ℤ\nh : -x < ↑n\n⊢ -↑n < x",
"state_before": "α : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nn : ℤ\nh : -x < ↑n\n⊢ ↑(-n) < x",
"tactic": "rw [Int.cast_neg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : StrictOrderedRing α\ninst✝ : Archimedean α\nx : α\nn : ℤ\nh : -x < ↑n\n⊢ -↑n < x",
"tactic": "exact neg_lt.1 h"
}
] |
[
156,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Algebra/Hom/Aut.lean
|
AddAut.one_apply
|
[] |
[
226,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
isIntegral_sub
|
[] |
[
518,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.not_sqLe_succ
|
[] |
[
872,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
871,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.mapQ_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.264287\nM : Type ?u.264290\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.265303\nM₂ : Type ?u.265306\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\n⊢ p ≤ comap 0 q",
"tactic": "simp"
},
{
"state_after": "case h.h\nR : Type u_2\nM : Type u_1\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type u_3\nM₂ : Type u_4\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nh : optParam (p ≤ comap 0 q) (_ : p ≤ comap 0 q)\nx✝ : M\n⊢ ↑(comp (mapQ p q 0 h) (mkQ p)) x✝ = ↑(comp 0 (mkQ p)) x✝",
"state_before": "R : Type u_2\nM : Type u_1\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type u_3\nM₂ : Type u_4\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nh : optParam (p ≤ comap 0 q) (_ : p ≤ comap 0 q)\n⊢ mapQ p q 0 h = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h\nR : Type u_2\nM : Type u_1\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type u_3\nM₂ : Type u_4\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nq : Submodule R₂ M₂\nh : optParam (p ≤ comap 0 q) (_ : p ≤ comap 0 q)\nx✝ : M\n⊢ ↑(comp (mapQ p q 0 h) (mkQ p)) x✝ = ↑(comp 0 (mkQ p)) x✝",
"tactic": "simp"
}
] |
[
426,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.HasColimit.isoOfEquivalence_inv_π
|
[
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝⁴ : Category J\nK : Type u₂\ninst✝³ : Category K\nC : Type u\ninst✝² : Category C\nF✝ F : J ⥤ C\ninst✝¹ : HasColimit F\nG : K ⥤ C\ninst✝ : HasColimit G\ne : J ≌ K\nw : e.functor ⋙ G ≅ F\nk : K\n⊢ colimit.ι G k ≫ (isoOfEquivalence e w).inv =\n G.map ((Equivalence.counitInv e).app k) ≫ w.hom.app (e.inverse.obj k) ≫ colimit.ι F (e.inverse.obj k)",
"tactic": "simp [HasColimit.isoOfEquivalence, IsColimit.coconePointsIsoOfEquivalence_inv]"
}
] |
[
960,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
956,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.mem_image
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.61453\ninst✝ : DecidableEq β\nf g : α → β\ns : Finset α\nt : Finset β\na : α\nb c : β\n⊢ b ∈ image f s ↔ ∃ a, a ∈ s ∧ f a = b",
"tactic": "simp only [mem_def, image_val, mem_dedup, Multiset.mem_map, exists_prop]"
}
] |
[
328,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.IsNormal.blsub_eq
|
[
{
"state_after": "α : Type ?u.366269\nβ : Type ?u.366272\nγ : Type ?u.366275\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\nH : IsNormal f\no : Ordinal\nh : IsLimit o\n⊢ ∀ (a : Ordinal), a < o → f a < f (succ a)",
"state_before": "α : Type ?u.366269\nβ : Type ?u.366272\nγ : Type ?u.366275\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\nH : IsNormal f\no : Ordinal\nh : IsLimit o\n⊢ (blsub o fun x x_1 => f x) = f o",
"tactic": "rw [← IsNormal.bsup_eq.{u, v} H h, bsup_eq_blsub_of_lt_succ_limit h]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.366269\nβ : Type ?u.366272\nγ : Type ?u.366275\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\nH : IsNormal f\no : Ordinal\nh : IsLimit o\n⊢ ∀ (a : Ordinal), a < o → f a < f (succ a)",
"tactic": "exact fun a _ => H.1 a"
}
] |
[
1970,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1967,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
Measurable.subtype_coe
|
[] |
[
564,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
562,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.swap_mul_self
|
[] |
[
519,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
518,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean
|
SimpleGraph.chromaticNumber_bddBelow
|
[] |
[
250,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/SetTheory/Ordinal/Principal.lean
|
Ordinal.principal_add_isLimit
|
[
{
"state_after": "case refine'_1\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\nho₀ : o = 0\n⊢ False\n\ncase refine'_2\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\na : Ordinal\nhao : a < o\n⊢ succ a < o",
"state_before": "o : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\n⊢ IsLimit o",
"tactic": "refine' ⟨fun ho₀ => _, fun a hao => _⟩"
},
{
"state_after": "case refine'_1\no : Ordinal\nho₁ : 1 < 0\nho : Principal (fun x x_1 => x + x_1) o\nho₀ : o = 0\n⊢ False",
"state_before": "case refine'_1\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\nho₀ : o = 0\n⊢ False",
"tactic": "rw [ho₀] at ho₁"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\no : Ordinal\nho₁ : 1 < 0\nho : Principal (fun x x_1 => x + x_1) o\nho₀ : o = 0\n⊢ False",
"tactic": "exact not_lt_of_gt zero_lt_one ho₁"
},
{
"state_after": "case refine'_2.inl\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\na : Ordinal\nhao : a < o\nha : a = 0\n⊢ succ a < o\n\ncase refine'_2.inr\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\na : Ordinal\nhao : a < o\nha : a ≠ 0\n⊢ succ a < o",
"state_before": "case refine'_2\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\na : Ordinal\nhao : a < o\n⊢ succ a < o",
"tactic": "cases' eq_or_ne a 0 with ha ha"
},
{
"state_after": "case refine'_2.inl\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\na : Ordinal\nhao : a < o\nha : a = 0\n⊢ 1 < o",
"state_before": "case refine'_2.inl\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\na : Ordinal\nhao : a < o\nha : a = 0\n⊢ succ a < o",
"tactic": "rw [ha, succ_zero]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.inl\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\na : Ordinal\nhao : a < o\nha : a = 0\n⊢ 1 < o",
"tactic": "exact ho₁"
},
{
"state_after": "case refine'_2.inr\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\na : Ordinal\nhao : a < o\nha : a ≠ 0\n⊢ succ a ≤ (fun x x_1 => x + x_1) a a",
"state_before": "case refine'_2.inr\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\na : Ordinal\nhao : a < o\nha : a ≠ 0\n⊢ succ a < o",
"tactic": "refine' lt_of_le_of_lt _ (ho hao hao)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.inr\no : Ordinal\nho₁ : 1 < o\nho : Principal (fun x x_1 => x + x_1) o\na : Ordinal\nhao : a < o\nha : a ≠ 0\n⊢ succ a ≤ (fun x x_1 => x + x_1) a a",
"tactic": "rwa [← add_one_eq_succ, add_le_add_iff_left, one_le_iff_ne_zero]"
}
] |
[
152,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.op_smul_finset_smul_eq_smul_smul_finset
|
[
{
"state_after": "case a\nF : Type ?u.755717\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\ninst✝⁴ : DecidableEq β\ninst✝³ : DecidableEq γ\ninst✝² : SMul αᵐᵒᵖ β\ninst✝¹ : SMul β γ\ninst✝ : SMul α γ\na : α\ns : Finset β\nt : Finset γ\nh : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c\na✝ : γ\n⊢ a✝ ∈ (op a • s) • t ↔ a✝ ∈ s • a • t",
"state_before": "F : Type ?u.755717\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\ninst✝⁴ : DecidableEq β\ninst✝³ : DecidableEq γ\ninst✝² : SMul αᵐᵒᵖ β\ninst✝¹ : SMul β γ\ninst✝ : SMul α γ\na : α\ns : Finset β\nt : Finset γ\nh : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c\n⊢ (op a • s) • t = s • a • t",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nF : Type ?u.755717\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\ninst✝⁴ : DecidableEq β\ninst✝³ : DecidableEq γ\ninst✝² : SMul αᵐᵒᵖ β\ninst✝¹ : SMul β γ\ninst✝ : SMul α γ\na : α\ns : Finset β\nt : Finset γ\nh : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c\na✝ : γ\n⊢ a✝ ∈ (op a • s) • t ↔ a✝ ∈ s • a • t",
"tactic": "simp [mem_smul, mem_smul_finset, h]"
}
] |
[
1826,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1823,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.Insert.comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.108268\nγ : Type ?u.108271\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na✝ b✝ a b : α\ns : Finset α\nx : α\n⊢ x ∈ insert a (insert b s) ↔ x ∈ insert b (insert a s)",
"tactic": "simp only [mem_insert, or_left_comm]"
}
] |
[
1117,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1116,
1
] |
Mathlib/Data/Sym/Basic.lean
|
Sym.attach_nil
|
[] |
[
452,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.AbsTheory.abs_re_le_abs
|
[
{
"state_after": "z : ℂ\n⊢ z.re * z.re ≤ ↑normSq z",
"state_before": "z : ℂ\n⊢ abs z.re ≤ Real.sqrt (↑normSq z)",
"tactic": "rw [mul_self_le_mul_self_iff (abs_nonneg z.re) (abs_nonneg' _), abs_mul_abs_self, mul_self_abs]"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\n⊢ z.re * z.re ≤ ↑normSq z",
"tactic": "apply re_sq_le_normSq"
}
] |
[
918,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
916,
9
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.isNilpotent_of_top_iff
|
[] |
[
504,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Algebra/MonoidAlgebra/Support.lean
|
AddMonoidAlgebra.mem_span_support
|
[
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\ninst✝¹ : Semiring k\ninst✝ : AddZeroClass G\nf : AddMonoidAlgebra k G\n⊢ f ∈ Submodule.span k (↑(of k G) '' ↑f.support)",
"tactic": "erw [of, MonoidHom.coe_mk, ← Finsupp.supported_eq_span_single, Finsupp.mem_supported]"
}
] |
[
145,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Analysis/NormedSpace/MStructure.lean
|
IsLprojection.Lcomplement_iff
|
[] |
[
105,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
|
CategoryTheory.Limits.pullback_diagonal_map_snd_fst_fst
|
[
{
"state_after": "no goals",
"state_before": "C : Type ?u.9333\ninst✝¹ : Category C\nX Y Z : C\ninst✝ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\n⊢ (i₁ ≫ snd) ≫ i = (i₁ ≫ fst) ≫ f",
"tactic": "simp [condition]"
},
{
"state_after": "no goals",
"state_before": "C : Type ?u.9333\ninst✝¹ : Category C\nX Y Z : C\ninst✝ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\n⊢ (i₂ ≫ snd) ≫ i = (i₂ ≫ fst) ≫ f",
"tactic": "simp [condition]"
},
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\nX Y Z : C\ninst✝ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\n⊢ snd ≫ fst ≫ i₁ ≫ fst = fst ≫ 𝟙 X",
"state_before": "C : Type u_2\ninst✝¹ : Category C\nX Y Z : C\ninst✝ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\n⊢ snd ≫ fst ≫ i₁ ≫ fst = fst",
"tactic": "conv_rhs => rw [← Category.comp_id pullback.fst]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\nX Y Z : C\ninst✝ : HasPullbacks C\nU V₁ V₂ : C\nf : X ⟶ Y\ni : U ⟶ Y\ni₁ : V₁ ⟶ pullback f i\ni₂ : V₂ ⟶ pullback f i\n⊢ snd ≫ fst ≫ i₁ ≫ fst = fst ≫ 𝟙 X",
"tactic": "rw [← diagonal_fst f, pullback.condition_assoc, pullback.lift_fst]"
}
] |
[
100,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.coe_restrictScalars
|
[] |
[
1680,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1678,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Matrix.mulVec_stdBasis_apply
|
[] |
[
273,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/Algebra/Homology/ImageToKernel.lean
|
imageToKernel_comp_mono
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.14901\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝² : HasKernels V\ninst✝¹ : HasImages V\nD : V\nh : C ⟶ D\ninst✝ : Mono h\nw : f ≫ g ≫ h = 0\n⊢ (f ≫ g) ≫ h = 0 ≫ h",
"tactic": "simpa using w"
},
{
"state_after": "case h\nι : Type ?u.14901\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝² : HasKernels V\ninst✝¹ : HasImages V\nD : V\nh : C ⟶ D\ninst✝ : Mono h\nw : f ≫ g ≫ h = 0\n⊢ imageToKernel f (g ≫ h) w ≫ Subobject.arrow (kernelSubobject (g ≫ h)) =\n (imageToKernel f g (_ : f ≫ g = 0) ≫\n (Subobject.isoOfEq (kernelSubobject (g ≫ h)) (kernelSubobject g)\n (_ : kernelSubobject (g ≫ h) = kernelSubobject g)).inv) ≫\n Subobject.arrow (kernelSubobject (g ≫ h))",
"state_before": "ι : Type ?u.14901\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝² : HasKernels V\ninst✝¹ : HasImages V\nD : V\nh : C ⟶ D\ninst✝ : Mono h\nw : f ≫ g ≫ h = 0\n⊢ imageToKernel f (g ≫ h) w =\n imageToKernel f g (_ : f ≫ g = 0) ≫\n (Subobject.isoOfEq (kernelSubobject (g ≫ h)) (kernelSubobject g)\n (_ : kernelSubobject (g ≫ h) = kernelSubobject g)).inv",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type ?u.14901\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nA B C : V\nf : A ⟶ B\ng : B ⟶ C\ninst✝² : HasKernels V\ninst✝¹ : HasImages V\nD : V\nh : C ⟶ D\ninst✝ : Mono h\nw : f ≫ g ≫ h = 0\n⊢ imageToKernel f (g ≫ h) w ≫ Subobject.arrow (kernelSubobject (g ≫ h)) =\n (imageToKernel f g (_ : f ≫ g = 0) ≫\n (Subobject.isoOfEq (kernelSubobject (g ≫ h)) (kernelSubobject g)\n (_ : kernelSubobject (g ≫ h) = kernelSubobject g)).inv) ≫\n Subobject.arrow (kernelSubobject (g ≫ h))",
"tactic": "simp"
}
] |
[
122,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/FieldTheory/Separable.lean
|
Polynomial.multiplicity_le_one_of_separable
|
[
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n⊢ IsUnit q",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhq : ¬IsUnit q\nhsep : Separable p\n⊢ multiplicity q p ≤ 1",
"tactic": "contrapose! hq"
},
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n⊢ q * q ∣ p",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n⊢ IsUnit q",
"tactic": "apply isUnit_of_self_mul_dvd_separable hsep"
},
{
"state_after": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n⊢ q ^ 2 ∣ p",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n⊢ q * q ∣ p",
"tactic": "rw [← sq]"
},
{
"state_after": "case a\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n⊢ ↑2 ≤ multiplicity q p",
"state_before": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n⊢ q ^ 2 ∣ p",
"tactic": "apply multiplicity.pow_dvd_of_le_multiplicity"
},
{
"state_after": "case a\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\nh : { Dom := 1.Dom ∧ 1.Dom, get := fun x => 2 } ≤ multiplicity q p\n⊢ ↑2 ≤ multiplicity q p",
"state_before": "case a\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\n⊢ ↑2 ≤ multiplicity q p",
"tactic": "have h : ⟨Part.Dom 1 ∧ Part.Dom 1, fun _ ↦ 2⟩ ≤ multiplicity q p := PartENat.add_one_le_of_lt hq"
},
{
"state_after": "case a\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\nh : { Dom := 1.Dom, get := fun x => 2 } ≤ multiplicity q p\n⊢ ↑2 ≤ multiplicity q p",
"state_before": "case a\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\nh : { Dom := 1.Dom ∧ 1.Dom, get := fun x => 2 } ≤ multiplicity q p\n⊢ ↑2 ≤ multiplicity q p",
"tactic": "rw [and_self] at h"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np✝ q✝ : ℕ\np q : R[X]\nhsep : Separable p\nhq : 1 < multiplicity q p\nh : { Dom := 1.Dom, get := fun x => 2 } ≤ multiplicity q p\n⊢ ↑2 ≤ multiplicity q p",
"tactic": "exact h"
}
] |
[
160,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.orthogonalProjection_vsub_mem_direction_orthogonal
|
[] |
[
432,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.sdiff_sdiff_eq_sdiff_union
|
[] |
[
2303,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2302,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.get_rowLens
|
[
{
"state_after": "no goals",
"state_before": "μ : YoungDiagram\ni : Fin (List.length (rowLens μ))\n⊢ List.get (rowLens μ) i = rowLen μ ↑i",
"tactic": "simp only [rowLens, List.get_range, List.get_map]"
}
] |
[
421,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffWithinAt.prod_map'
|
[] |
[
1597,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1593,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.mk'_mem_incidenceSet_iff
|
[] |
[
835,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
834,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
Prod.fst_mul_snd
|
[] |
[
128,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.map_add_left_Ioc
|
[
{
"state_after": "ι : Type ?u.221440\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ ↑(addLeftEmbedding c) '' Set.Ioc a b = Set.Ioc (c + a) (c + b)",
"state_before": "ι : Type ?u.221440\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (addLeftEmbedding c) (Ioc a b) = Ioc (c + a) (c + b)",
"tactic": "rw [← coe_inj, coe_map, coe_Ioc, coe_Ioc]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.221440\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ ↑(addLeftEmbedding c) '' Set.Ioc a b = Set.Ioc (c + a) (c + b)",
"tactic": "exact Set.image_const_add_Ioc _ _ _"
}
] |
[
1072,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1069,
1
] |
Mathlib/RingTheory/Polynomial/Pochhammer.lean
|
pochhammer_eval_cast
|
[
{
"state_after": "no goals",
"state_before": "S : Type u\ninst✝ : Semiring S\nn k : ℕ\n⊢ ↑(eval k (pochhammer ℕ n)) = eval (↑k) (pochhammer S n)",
"tactic": "rw [← pochhammer_map (algebraMap ℕ S), eval_map, ← eq_natCast (algebraMap ℕ S), eval₂_at_nat_cast,\n Nat.cast_id, eq_natCast]"
}
] |
[
82,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/CategoryTheory/Preadditive/Mat.lean
|
CategoryTheory.Mat.comp_def
|
[] |
[
609,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
607,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.ofReal_intCast
|
[] |
[
665,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
Mathlib/RingTheory/NonZeroDivisors.lean
|
mul_left_coe_nonZeroDivisors_eq_zero_iff
|
[] |
[
65,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Std/Data/Int/DivMod.lean
|
Int.div_eq_ediv_of_dvd
|
[
{
"state_after": "no goals",
"state_before": "a b : Int\nh : b ∣ a\n⊢ div a b = a / b",
"tactic": "if b0 : b = 0 then simp [b0]\nelse rw [Int.div_eq_iff_eq_mul_left b0 h, ← Int.ediv_eq_iff_eq_mul_left b0 h]"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh : b ∣ a\nb0 : b = 0\n⊢ div a b = a / b",
"tactic": "simp [b0]"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh : b ∣ a\nb0 : ¬b = 0\n⊢ div a b = a / b",
"tactic": "rw [Int.div_eq_iff_eq_mul_left b0 h, ← Int.ediv_eq_iff_eq_mul_left b0 h]"
}
] |
[
795,
80
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
793,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean
|
PiNat.cylinder_eq_pi
|
[
{
"state_after": "case h\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\n⊢ y ∈ cylinder x n ↔ y ∈ Set.pi ↑(Finset.range n) fun i => {x i}",
"state_before": "E : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\n⊢ cylinder x n = Set.pi ↑(Finset.range n) fun i => {x i}",
"tactic": "ext y"
},
{
"state_after": "no goals",
"state_before": "case h\nE : ℕ → Type u_1\nx : (n : ℕ) → E n\nn : ℕ\ny : (n : ℕ) → E n\n⊢ y ∈ cylinder x n ↔ y ∈ Set.pi ↑(Finset.range n) fun i => {x i}",
"tactic": "simp [cylinder]"
}
] |
[
117,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Analysis/Convex/Combination.lean
|
convexHull_prod
|
[] |
[
426,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEStronglyMeasurable.smul_const
|
[] |
[
1344,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1342,
11
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
Pmf.toOuterMeasure_apply_eq_zero_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.25862\nγ : Type ?u.25865\np : Pmf α\ns t : Set α\n⊢ (∀ (i : α), Set.indicator s (↑p) i = 0) ↔ Disjoint (support p) s",
"state_before": "α : Type u_1\nβ : Type ?u.25862\nγ : Type ?u.25865\np : Pmf α\ns t : Set α\n⊢ ↑(toOuterMeasure p) s = 0 ↔ Disjoint (support p) s",
"tactic": "rw [toOuterMeasure_apply, ENNReal.tsum_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.25862\nγ : Type ?u.25865\np : Pmf α\ns t : Set α\n⊢ (∀ (i : α), Set.indicator s (↑p) i = 0) ↔ Disjoint (support p) s",
"tactic": "exact Function.funext_iff.symm.trans Set.indicator_eq_zero'"
}
] |
[
197,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/Int/Cast/Lemmas.lean
|
Int.cast_le
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.15849\nι : Type ?u.15852\nα : Type u_1\nβ : Type ?u.15858\ninst✝¹ : OrderedRing α\ninst✝ : Nontrivial α\nm n : ℤ\n⊢ ↑m ≤ ↑n ↔ m ≤ n",
"tactic": "rw [← sub_nonneg, ← cast_sub, cast_nonneg, sub_nonneg]"
}
] |
[
132,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.NullMeasurableSet.const
|
[] |
[
217,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
11
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.toLinearMap_symm_comp_eq
|
[
{
"state_after": "case mp.h\nR : Type ?u.209595\nR₁ : Type u_2\nR₂ : Type u_5\nR₃ : Type u_1\nk : Type ?u.209607\nS : Type ?u.209610\nM : Type ?u.209613\nM₁ : Type u_4\nM₂ : Type u_6\nM₃ : Type u_3\nN₁ : Type ?u.209625\nN₂ : Type ?u.209628\nN₃ : Type ?u.209631\nN₄ : Type ?u.209634\nι : Type ?u.209637\nM₄ : Type ?u.209640\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₃ →ₛₗ[σ₃₁] M₁\ng : M₃ →ₛₗ[σ₃₂] M₂\nH : LinearMap.comp (↑(symm e₁₂)) g = f\nx✝ : M₃\n⊢ ↑g x✝ = ↑(LinearMap.comp (↑e₁₂) f) x✝\n\ncase mpr.h\nR : Type ?u.209595\nR₁ : Type u_2\nR₂ : Type u_5\nR₃ : Type u_1\nk : Type ?u.209607\nS : Type ?u.209610\nM : Type ?u.209613\nM₁ : Type u_4\nM₂ : Type u_6\nM₃ : Type u_3\nN₁ : Type ?u.209625\nN₂ : Type ?u.209628\nN₃ : Type ?u.209631\nN₄ : Type ?u.209634\nι : Type ?u.209637\nM₄ : Type ?u.209640\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₃ →ₛₗ[σ₃₁] M₁\ng : M₃ →ₛₗ[σ₃₂] M₂\nH : g = LinearMap.comp (↑e₁₂) f\nx✝ : M₃\n⊢ ↑(LinearMap.comp (↑(symm e₁₂)) g) x✝ = ↑f x✝",
"state_before": "R : Type ?u.209595\nR₁ : Type u_2\nR₂ : Type u_5\nR₃ : Type u_1\nk : Type ?u.209607\nS : Type ?u.209610\nM : Type ?u.209613\nM₁ : Type u_4\nM₂ : Type u_6\nM₃ : Type u_3\nN₁ : Type ?u.209625\nN₂ : Type ?u.209628\nN₃ : Type ?u.209631\nN₄ : Type ?u.209634\nι : Type ?u.209637\nM₄ : Type ?u.209640\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₃ →ₛₗ[σ₃₁] M₁\ng : M₃ →ₛₗ[σ₃₂] M₂\n⊢ LinearMap.comp (↑(symm e₁₂)) g = f ↔ g = LinearMap.comp (↑e₁₂) f",
"tactic": "constructor <;> intro H <;> ext"
},
{
"state_after": "no goals",
"state_before": "case mp.h\nR : Type ?u.209595\nR₁ : Type u_2\nR₂ : Type u_5\nR₃ : Type u_1\nk : Type ?u.209607\nS : Type ?u.209610\nM : Type ?u.209613\nM₁ : Type u_4\nM₂ : Type u_6\nM₃ : Type u_3\nN₁ : Type ?u.209625\nN₂ : Type ?u.209628\nN₃ : Type ?u.209631\nN₄ : Type ?u.209634\nι : Type ?u.209637\nM₄ : Type ?u.209640\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₃ →ₛₗ[σ₃₁] M₁\ng : M₃ →ₛₗ[σ₃₂] M₂\nH : LinearMap.comp (↑(symm e₁₂)) g = f\nx✝ : M₃\n⊢ ↑g x✝ = ↑(LinearMap.comp (↑e₁₂) f) x✝",
"tactic": "simp [← H, ← e₁₂.toEquiv.symm_comp_eq f g]"
},
{
"state_after": "no goals",
"state_before": "case mpr.h\nR : Type ?u.209595\nR₁ : Type u_2\nR₂ : Type u_5\nR₃ : Type u_1\nk : Type ?u.209607\nS : Type ?u.209610\nM : Type ?u.209613\nM₁ : Type u_4\nM₂ : Type u_6\nM₃ : Type u_3\nN₁ : Type ?u.209625\nN₂ : Type ?u.209628\nN₃ : Type ?u.209631\nN₄ : Type ?u.209634\nι : Type ?u.209637\nM₄ : Type ?u.209640\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₃ →ₛₗ[σ₃₁] M₁\ng : M₃ →ₛₗ[σ₃₂] M₂\nH : g = LinearMap.comp (↑e₁₂) f\nx✝ : M₃\n⊢ ↑(LinearMap.comp (↑(symm e₁₂)) g) x✝ = ↑f x✝",
"tactic": "simp [H, e₁₂.toEquiv.symm_comp_eq f g]"
}
] |
[
457,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Mathlib/CategoryTheory/Whiskering.lean
|
CategoryTheory.isoWhiskerRight_inv
|
[] |
[
189,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.update_eq_single_add_erase
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.295412\nγ : Type ?u.295415\nι : Type ?u.295418\nM : Type u_2\nM' : Type ?u.295424\nN : Type ?u.295427\nP : Type ?u.295430\nG : Type ?u.295433\nH : Type ?u.295436\nR : Type ?u.295439\nS : Type ?u.295442\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\nj : α\n⊢ ↑(update f a b) j = ↑(single a b + erase a f) j",
"state_before": "α : Type u_1\nβ : Type ?u.295412\nγ : Type ?u.295415\nι : Type ?u.295418\nM : Type u_2\nM' : Type ?u.295424\nN : Type ?u.295427\nP : Type ?u.295430\nG : Type ?u.295433\nH : Type ?u.295436\nR : Type ?u.295439\nS : Type ?u.295442\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\n⊢ update f a b = single a b + erase a f",
"tactic": "ext j"
},
{
"state_after": "case h.inl\nα : Type u_1\nβ : Type ?u.295412\nγ : Type ?u.295415\nι : Type ?u.295418\nM : Type u_2\nM' : Type ?u.295424\nN : Type ?u.295427\nP : Type ?u.295430\nG : Type ?u.295433\nH : Type ?u.295436\nR : Type ?u.295439\nS : Type ?u.295442\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\n⊢ ↑(update f a b) a = ↑(single a b + erase a f) a\n\ncase h.inr\nα : Type u_1\nβ : Type ?u.295412\nγ : Type ?u.295415\nι : Type ?u.295418\nM : Type u_2\nM' : Type ?u.295424\nN : Type ?u.295427\nP : Type ?u.295430\nG : Type ?u.295433\nH : Type ?u.295436\nR : Type ?u.295439\nS : Type ?u.295442\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\nj : α\nh : a ≠ j\n⊢ ↑(update f a b) j = ↑(single a b + erase a f) j",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.295412\nγ : Type ?u.295415\nι : Type ?u.295418\nM : Type u_2\nM' : Type ?u.295424\nN : Type ?u.295427\nP : Type ?u.295430\nG : Type ?u.295433\nH : Type ?u.295436\nR : Type ?u.295439\nS : Type ?u.295442\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\nj : α\n⊢ ↑(update f a b) j = ↑(single a b + erase a f) j",
"tactic": "rcases eq_or_ne a j with (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nα : Type u_1\nβ : Type ?u.295412\nγ : Type ?u.295415\nι : Type ?u.295418\nM : Type u_2\nM' : Type ?u.295424\nN : Type ?u.295427\nP : Type ?u.295430\nG : Type ?u.295433\nH : Type ?u.295436\nR : Type ?u.295439\nS : Type ?u.295442\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\n⊢ ↑(update f a b) a = ↑(single a b + erase a f) a",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nα : Type u_1\nβ : Type ?u.295412\nγ : Type ?u.295415\nι : Type ?u.295418\nM : Type u_2\nM' : Type ?u.295424\nN : Type ?u.295427\nP : Type ?u.295430\nG : Type ?u.295433\nH : Type ?u.295436\nR : Type ?u.295439\nS : Type ?u.295442\ninst✝ : AddZeroClass M\nf : α →₀ M\na : α\nb : M\nj : α\nh : a ≠ j\n⊢ ↑(update f a b) j = ↑(single a b + erase a f) j",
"tactic": "simp [Function.update_noteq h.symm, single_apply, h, erase_ne, h.symm]"
}
] |
[
1047,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Order/CompleteBooleanAlgebra.lean
|
iSup₂_inf_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nκ : ι → Sort u_1\ninst✝ : Frame α\ns t : Set α\na✝ b : α\nf : (i : ι) → κ i → α\na : α\n⊢ (⨆ (i : ι) (j : κ i), f i j) ⊓ a = ⨆ (i : ι) (j : κ i), f i j ⊓ a",
"tactic": "simp only [iSup_inf_eq]"
}
] |
[
110,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Order/SymmDiff.lean
|
sdiff_symmDiff'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.61584\nα : Type u_1\nβ : Type ?u.61590\nπ : ι → Type ?u.61595\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ c \\ a ∆ b = c ⊓ a ⊓ b ⊔ c \\ (a ⊔ b)",
"tactic": "rw [sdiff_symmDiff, sdiff_sup, sup_comm]"
}
] |
[
420,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffAt.inv
|
[] |
[
1711,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1709,
8
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
IsFractional.div_of_nonzero
|
[
{
"state_after": "case intro.intro\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\n⊢ IsFractional R₁⁰ (I / J)",
"state_before": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\n⊢ IsFractional R₁⁰ (I / J)",
"tactic": "obtain ⟨y, mem_J, not_mem_zero⟩ :=\n SetLike.exists_of_lt (show 0 < J by simpa only using bot_lt_iff_ne_bot.mpr h)"
},
{
"state_after": "case intro.intro.intro\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\n⊢ IsFractional R₁⁰ (I / J)",
"state_before": "case intro.intro\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\n⊢ IsFractional R₁⁰ (I / J)",
"tactic": "obtain ⟨y', hy'⟩ := hJ y mem_J"
},
{
"state_after": "case intro.intro.intro\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\n⊢ aI * y' ∈ R₁⁰ ∧ ∀ (b : K), b ∈ I / J → IsInteger R₁ ((aI * y') • b)",
"state_before": "case intro.intro.intro\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\n⊢ IsFractional R₁⁰ (I / J)",
"tactic": "use aI * y'"
},
{
"state_after": "case intro.intro.intro.left\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\n⊢ aI * y' ∈ R₁⁰\n\ncase intro.intro.intro.right\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\n⊢ ∀ (b : K), b ∈ I / J → IsInteger R₁ ((aI * y') • b)",
"state_before": "case intro.intro.intro\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\n⊢ aI * y' ∈ R₁⁰ ∧ ∀ (b : K), b ∈ I / J → IsInteger R₁ ((aI * y') • b)",
"tactic": "constructor"
},
{
"state_after": "case intro.intro.intro.right\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\nb : K\nhb : b ∈ I / J\n⊢ IsInteger R₁ ((aI * y') • b)",
"state_before": "case intro.intro.intro.right\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\n⊢ ∀ (b : K), b ∈ I / J → IsInteger R₁ ((aI * y') • b)",
"tactic": "intro b hb"
},
{
"state_after": "case h.e'_6\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\nb : K\nhb : b ∈ I / J\n⊢ (aI * y') • b = aI • (b * aJ • y)",
"state_before": "case intro.intro.intro.right\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\nb : K\nhb : b ∈ I / J\n⊢ IsInteger R₁ ((aI * y') • b)",
"tactic": "convert hI _ (hb _ (Submodule.smul_mem _ aJ mem_J)) using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\nb : K\nhb : b ∈ I / J\n⊢ (aI * y') • b = aI • (b * aJ • y)",
"tactic": "rw [← hy', mul_comm b, ← Algebra.smul_def, mul_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\n⊢ 0 < J",
"tactic": "simpa only using bot_lt_iff_ne_bot.mpr h"
},
{
"state_after": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\n⊢ y' ≠ 0",
"state_before": "case intro.intro.intro.left\nR : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\n⊢ aI * y' ∈ R₁⁰",
"tactic": "apply (nonZeroDivisors R₁).mul_mem haI (mem_nonZeroDivisors_iff_ne_zero.mpr _)"
},
{
"state_after": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\ny'_eq_zero : y' = 0\n⊢ False",
"state_before": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\n⊢ y' ≠ 0",
"tactic": "intro y'_eq_zero"
},
{
"state_after": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\ny'_eq_zero : y' = 0\nthis : ↑(algebraMap R₁ K) aJ * y = 0\n⊢ False",
"state_before": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\ny'_eq_zero : y' = 0\n⊢ False",
"tactic": "have : algebraMap R₁ K aJ * y = 0 := by\n rw [← Algebra.smul_def, ← hy', y'_eq_zero, RingHom.map_zero]"
},
{
"state_after": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\ny'_eq_zero : y' = 0\nthis : ↑(algebraMap R₁ K) aJ * y = 0\ny_zero : y = 0\n⊢ False",
"state_before": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\ny'_eq_zero : y' = 0\nthis : ↑(algebraMap R₁ K) aJ * y = 0\n⊢ False",
"tactic": "have y_zero :=\n (mul_eq_zero.mp this).resolve_left\n (mt ((injective_iff_map_eq_zero (algebraMap R₁ K)).1 (IsFractionRing.injective _ _) _)\n (mem_nonZeroDivisors_iff_ne_zero.mp haJ))"
},
{
"state_after": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\ny'_eq_zero : y' = 0\nthis : ↑(algebraMap R₁ K) aJ * y = 0\ny_zero : y = 0\n⊢ y ∈ 0",
"state_before": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\ny'_eq_zero : y' = 0\nthis : ↑(algebraMap R₁ K) aJ * y = 0\ny_zero : y = 0\n⊢ False",
"tactic": "apply not_mem_zero"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\ny'_eq_zero : y' = 0\nthis : ↑(algebraMap R₁ K) aJ * y = 0\ny_zero : y = 0\n⊢ y ∈ 0",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1138684\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type ?u.1138891\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nR₁ : Type u_1\ninst✝³ : CommRing R₁\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R₁ K\nfrac : IsFractionRing R₁ K\ninst✝ : IsDomain R₁\nI J : Submodule R₁ K\naI : R₁\nhaI : aI ∈ R₁⁰\nhI : ∀ (b : K), b ∈ I → IsInteger R₁ (aI • b)\naJ : R₁\nhaJ : aJ ∈ R₁⁰\nhJ : ∀ (b : K), b ∈ J → IsInteger R₁ (aJ • b)\nh : J ≠ 0\ny : K\nmem_J : y ∈ J\nnot_mem_zero : ¬y ∈ 0\ny' : R₁\nhy' : ↑(algebraMap R₁ K) y' = aJ • y\ny'_eq_zero : y' = 0\n⊢ ↑(algebraMap R₁ K) aJ * y = 0",
"tactic": "rw [← Algebra.smul_def, ← hy', y'_eq_zero, RingHom.map_zero]"
}
] |
[
1086,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1066,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
tendstoUniformlyOn_iff_tendsto
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF✝ : ι → α → β\nf✝ : α → β\ns✝ s' : Set α\nx : α\np✝ : Filter ι\np' : Filter α\ng : ι → α\nF : ι → α → β\nf : α → β\np : Filter ι\ns : Set α\n⊢ TendstoUniformlyOn F f p s ↔ Tendsto (fun q => (f q.snd, F q.fst q.snd)) (p ×ˢ 𝓟 s) (𝓤 β)",
"tactic": "simp [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter, tendstoUniformlyOnFilter_iff_tendsto]"
}
] |
[
129,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Data/Set/Intervals/OrderIso.lean
|
OrderIso.preimage_Iio
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\nb : β\nx : α\n⊢ x ∈ ↑e ⁻¹' Iio b ↔ x ∈ Iio (↑(symm e) b)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\nb : β\n⊢ ↑e ⁻¹' Iio b = Iio (↑(symm e) b)",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\nb : β\nx : α\n⊢ x ∈ ↑e ⁻¹' Iio b ↔ x ∈ Iio (↑(symm e) b)",
"tactic": "simp [← e.lt_iff_lt]"
}
] |
[
40,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/Init/Data/Nat/Basic.lean
|
Nat.bit0_ne_zero
|
[] |
[
35,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
11
] |
Std/Data/RBMap/Alter.lean
|
Std.RBNode.Ordered.zoom'
|
[] |
[
287,
38
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
282,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.mul_inv_eq_iff_eq_mul_of_invertible
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.235209\nm : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA✝ B✝ A B C : Matrix n n α\ninst✝ : Invertible A\nh : B ⬝ A⁻¹ = C\n⊢ B = C ⬝ A",
"tactic": "rw [← h, inv_mul_cancel_right_of_invertible]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.235209\nm : Type u\nn : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA✝ B✝ A B C : Matrix n n α\ninst✝ : Invertible A\nh : B = C ⬝ A\n⊢ B ⬝ A⁻¹ = C",
"tactic": "rw [h, mul_inv_cancel_right_of_invertible]"
}
] |
[
383,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
1
] |
Mathlib/Order/OrdContinuous.lean
|
RightOrdContinuous.map_sInf'
|
[] |
[
238,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iInf_false
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.105197\nβ₂ : Type ?u.105200\nγ : Type ?u.105203\nι : Sort ?u.105206\nι' : Sort ?u.105209\nκ : ι → Sort ?u.105214\nκ' : ι' → Sort ?u.105219\ninst✝ : CompleteLattice α\nf g s✝ t : ι → α\na b : α\ns : False → α\n⊢ iInf s = ⊤",
"tactic": "simp"
}
] |
[
1296,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1295,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.codisjoint_inf_left
|
[] |
[
563,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
11
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.div_subset_iff
|
[] |
[
656,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Antitone.leftLim_le
|
[] |
[
306,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Set.Infinite.exists_gt
|
[] |
[
840,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
839,
1
] |
Mathlib/Combinatorics/Pigeonhole.lean
|
Fintype.exists_le_card_fiber_of_nsmul_le_card
|
[] |
[
426,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/Analysis/SpecialFunctions/Bernstein.lean
|
bernsteinApproximation.apply
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nf : C(↑I, ℝ)\nx : ↑I\n⊢ ↑(bernsteinApproximation n f) x = ∑ k : Fin (n + 1), ↑f k/ₙ * ↑(bernstein n ↑k) x",
"tactic": "simp [bernsteinApproximation]"
}
] |
[
175,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Data/Semiquot.lean
|
Semiquot.map_def
|
[] |
[
154,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.continuousAt_angle
|
[
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nx : P × P × P\nhx12 : x.fst ≠ x.snd.fst\nhx32 : x.snd.snd ≠ x.snd.fst\nf : P × P × P → V × V := fun y => (y.fst -ᵥ y.snd.fst, y.snd.snd -ᵥ y.snd.fst)\n⊢ ContinuousAt (fun y => ∠ y.fst y.snd.fst y.snd.snd) x",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nx : P × P × P\nhx12 : x.fst ≠ x.snd.fst\nhx32 : x.snd.snd ≠ x.snd.fst\n⊢ ContinuousAt (fun y => ∠ y.fst y.snd.fst y.snd.snd) x",
"tactic": "let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)"
},
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nx : P × P × P\nhx12 : x.fst ≠ x.snd.fst\nhx32 : x.snd.snd ≠ x.snd.fst\nf : P × P × P → V × V := fun y => (y.fst -ᵥ y.snd.fst, y.snd.snd -ᵥ y.snd.fst)\nhf1 : (f x).fst ≠ 0\n⊢ ContinuousAt (fun y => ∠ y.fst y.snd.fst y.snd.snd) x",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nx : P × P × P\nhx12 : x.fst ≠ x.snd.fst\nhx32 : x.snd.snd ≠ x.snd.fst\nf : P × P × P → V × V := fun y => (y.fst -ᵥ y.snd.fst, y.snd.snd -ᵥ y.snd.fst)\n⊢ ContinuousAt (fun y => ∠ y.fst y.snd.fst y.snd.snd) x",
"tactic": "have hf1 : (f x).1 ≠ 0 := by simp [hx12]"
},
{
"state_after": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nx : P × P × P\nhx12 : x.fst ≠ x.snd.fst\nhx32 : x.snd.snd ≠ x.snd.fst\nf : P × P × P → V × V := fun y => (y.fst -ᵥ y.snd.fst, y.snd.snd -ᵥ y.snd.fst)\nhf1 : (f x).fst ≠ 0\nhf2 : (f x).snd ≠ 0\n⊢ ContinuousAt (fun y => ∠ y.fst y.snd.fst y.snd.snd) x",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nx : P × P × P\nhx12 : x.fst ≠ x.snd.fst\nhx32 : x.snd.snd ≠ x.snd.fst\nf : P × P × P → V × V := fun y => (y.fst -ᵥ y.snd.fst, y.snd.snd -ᵥ y.snd.fst)\nhf1 : (f x).fst ≠ 0\n⊢ ContinuousAt (fun y => ∠ y.fst y.snd.fst y.snd.snd) x",
"tactic": "have hf2 : (f x).2 ≠ 0 := by simp [hx32]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nx : P × P × P\nhx12 : x.fst ≠ x.snd.fst\nhx32 : x.snd.snd ≠ x.snd.fst\nf : P × P × P → V × V := fun y => (y.fst -ᵥ y.snd.fst, y.snd.snd -ᵥ y.snd.fst)\nhf1 : (f x).fst ≠ 0\nhf2 : (f x).snd ≠ 0\n⊢ ContinuousAt (fun y => ∠ y.fst y.snd.fst y.snd.snd) x",
"tactic": "exact (InnerProductGeometry.continuousAt_angle hf1 hf2).comp\n ((continuous_fst.vsub continuous_snd.fst).prod_mk\n (continuous_snd.snd.vsub continuous_snd.fst)).continuousAt"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nx : P × P × P\nhx12 : x.fst ≠ x.snd.fst\nhx32 : x.snd.snd ≠ x.snd.fst\nf : P × P × P → V × V := fun y => (y.fst -ᵥ y.snd.fst, y.snd.snd -ᵥ y.snd.fst)\n⊢ (f x).fst ≠ 0",
"tactic": "simp [hx12]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nx : P × P × P\nhx12 : x.fst ≠ x.snd.fst\nhx32 : x.snd.snd ≠ x.snd.fst\nf : P × P × P → V × V := fun y => (y.fst -ᵥ y.snd.fst, y.snd.snd -ᵥ y.snd.fst)\nhf1 : (f x).fst ≠ 0\n⊢ (f x).snd ≠ 0",
"tactic": "simp [hx32]"
}
] |
[
55,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/Algebra/Order/Group/MinMax.lean
|
max_one_div_max_inv_one_eq_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : Group α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ max a 1 / max a⁻¹ 1 = a",
"tactic": "rcases le_total a 1 with (h | h) <;> simp [h]"
}
] |
[
25,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
24,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.frequently_imp_distrib
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.165345\nι : Sort x\nf : Filter α\np q : α → Prop\n⊢ (∃ᶠ (x : α) in f, p x → q x) ↔ (∀ᶠ (x : α) in f, p x) → ∃ᶠ (x : α) in f, q x",
"tactic": "simp [imp_iff_not_or, not_eventually, frequently_or_distrib]"
}
] |
[
1356,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1354,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
Set.Countable.ae_not_mem
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.712897\nβ : Type ?u.712900\nγ : Type ?u.712903\nδ : Type ?u.712906\nι : Type ?u.712909\nR : Type ?u.712912\nR' : Type ?u.712915\nm0 : MeasurableSpace α✝\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α✝\ns✝ s' t : Set α✝\ninst✝¹ : NoAtoms μ✝\nα : Type u_1\nm : MeasurableSpace α\ns : Set α\nh : Set.Countable s\nμ : Measure α\ninst✝ : NoAtoms μ\n⊢ ∀ᵐ (x : α) ∂μ, ¬x ∈ s",
"tactic": "simpa only [ae_iff, Classical.not_not] using h.measure_zero μ"
}
] |
[
3310,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3308,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
StieltjesFunction.measure_Ioc
|
[
{
"state_after": "f : StieltjesFunction\na b : ℝ\n⊢ ↑↑{ toOuterMeasure := StieltjesFunction.outer f,\n m_iUnion :=\n (_ :\n ∀ (_s : ℕ → Set ℝ),\n (∀ (i : ℕ), MeasurableSet (_s i)) →\n Pairwise (Disjoint on _s) →\n ↑(StieltjesFunction.outer f) (⋃ (i : ℕ), _s i) = ∑' (i : ℕ), ↑(StieltjesFunction.outer f) (_s i)),\n trimmed := (_ : OuterMeasure.trim (StieltjesFunction.outer f) = StieltjesFunction.outer f) }\n (Ioc a b) =\n ofReal (↑f b - ↑f a)",
"state_before": "f : StieltjesFunction\na b : ℝ\n⊢ ↑↑(StieltjesFunction.measure f) (Ioc a b) = ofReal (↑f b - ↑f a)",
"tactic": "rw [StieltjesFunction.measure]"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\na b : ℝ\n⊢ ↑↑{ toOuterMeasure := StieltjesFunction.outer f,\n m_iUnion :=\n (_ :\n ∀ (_s : ℕ → Set ℝ),\n (∀ (i : ℕ), MeasurableSet (_s i)) →\n Pairwise (Disjoint on _s) →\n ↑(StieltjesFunction.outer f) (⋃ (i : ℕ), _s i) = ∑' (i : ℕ), ↑(StieltjesFunction.outer f) (_s i)),\n trimmed := (_ : OuterMeasure.trim (StieltjesFunction.outer f) = StieltjesFunction.outer f) }\n (Ioc a b) =\n ofReal (↑f b - ↑f a)",
"tactic": "exact f.outer_Ioc a b"
}
] |
[
517,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
padicNormE.eq_ratNorm
|
[] |
[
894,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
893,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
IsometryEquiv.mul_apply
|
[] |
[
555,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
StructureGroupoid.LocalInvariantProp.liftPropOn_chart
|
[] |
[
519,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
517,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.le_natTrailingDegree_mul
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\n⊢ natTrailingDegree p + natTrailingDegree q ≤ natTrailingDegree (p * q)",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\n⊢ natTrailingDegree p + natTrailingDegree q ≤ natTrailingDegree (p * q)",
"tactic": "have hp : p ≠ 0 := fun hp => h (by rw [hp, zero_mul])"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\nhq : q ≠ 0\n⊢ natTrailingDegree p + natTrailingDegree q ≤ natTrailingDegree (p * q)",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\n⊢ natTrailingDegree p + natTrailingDegree q ≤ natTrailingDegree (p * q)",
"tactic": "have hq : q ≠ 0 := fun hq => h (by rw [hq, mul_zero])"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\nhq : q ≠ 0\nthis : ∀ (p : R[X]), ↑(natTrailingDegree p) = ↑(natTrailingDegree p)\n⊢ natTrailingDegree p + natTrailingDegree q ≤ natTrailingDegree (p * q)",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\nhq : q ≠ 0\n⊢ natTrailingDegree p + natTrailingDegree q ≤ natTrailingDegree (p * q)",
"tactic": "have : ∀ (p : R[X]), WithTop.some (natTrailingDegree p) = Nat.cast (natTrailingDegree p) :=\n fun p ↦ rfl"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\nhq : q ≠ 0\nthis : ∀ (p : R[X]), ↑(natTrailingDegree p) = ↑(natTrailingDegree p)\n⊢ trailingDegree p + trailingDegree q ≤ trailingDegree (p * q)",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\nhq : q ≠ 0\nthis : ∀ (p : R[X]), ↑(natTrailingDegree p) = ↑(natTrailingDegree p)\n⊢ natTrailingDegree p + natTrailingDegree q ≤ natTrailingDegree (p * q)",
"tactic": "rw [← WithTop.coe_le_coe, WithTop.coe_add, this p, this q, this (p * q),\n ← trailingDegree_eq_natTrailingDegree hp, ← trailingDegree_eq_natTrailingDegree hq,\n ← trailingDegree_eq_natTrailingDegree h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\nhq : q ≠ 0\nthis : ∀ (p : R[X]), ↑(natTrailingDegree p) = ↑(natTrailingDegree p)\n⊢ trailingDegree p + trailingDegree q ≤ trailingDegree (p * q)",
"tactic": "exact le_trailingDegree_mul"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\nhp : p = 0\n⊢ p * q = 0",
"tactic": "rw [hp, zero_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nh : p * q ≠ 0\nhp : p ≠ 0\nhq : q = 0\n⊢ p * q = 0",
"tactic": "rw [hq, mul_zero]"
}
] |
[
380,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
1
] |
Mathlib/Algebra/GeomSum.lean
|
Odd.add_dvd_pow_add_pow
|
[
{
"state_after": "α : Type u\ninst✝ : CommRing α\nx y : α\nn : ℕ\nh : Odd n\nh₁ : (∑ i in range n, x ^ i * (-y) ^ (n - 1 - i)) * (x - -y) = x ^ n - (-y) ^ n\n⊢ x + y ∣ x ^ n + y ^ n",
"state_before": "α : Type u\ninst✝ : CommRing α\nx y : α\nn : ℕ\nh : Odd n\n⊢ x + y ∣ x ^ n + y ^ n",
"tactic": "have h₁ := geom_sum₂_mul x (-y) n"
},
{
"state_after": "α : Type u\ninst✝ : CommRing α\nx y : α\nn : ℕ\nh : Odd n\nh₁ : (∑ i in range n, x ^ i * (-y) ^ (n - 1 - i)) * (x + y) = x ^ n + y ^ n\n⊢ x + y ∣ x ^ n + y ^ n",
"state_before": "α : Type u\ninst✝ : CommRing α\nx y : α\nn : ℕ\nh : Odd n\nh₁ : (∑ i in range n, x ^ i * (-y) ^ (n - 1 - i)) * (x - -y) = x ^ n - (-y) ^ n\n⊢ x + y ∣ x ^ n + y ^ n",
"tactic": "rw [Odd.neg_pow h y, sub_neg_eq_add, sub_neg_eq_add] at h₁"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : CommRing α\nx y : α\nn : ℕ\nh : Odd n\nh₁ : (∑ i in range n, x ^ i * (-y) ^ (n - 1 - i)) * (x + y) = x ^ n + y ^ n\n⊢ x + y ∣ x ^ n + y ^ n",
"tactic": "exact Dvd.intro_left _ h₁"
}
] |
[
212,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Computability/Partrec.lean
|
Computable.to₂
|
[] |
[
297,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
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