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/Data/Set/Pointwise/Interval.lean
|
Set.image_sub_const_Ioc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x - a) '' Ioc b c = Ioc (b - a) (c - a)",
"tactic": "simp [sub_eq_neg_add]"
}
] |
[
401,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.tan_mul_I
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ tan (x * I) = tanh x * I",
"tactic": "rw [tan, sin_mul_I, cos_mul_I, mul_div_right_comm, tanh_eq_sinh_div_cosh]"
}
] |
[
843,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
842,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
Real.map_exp_nhds
|
[] |
[
354,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.neg_eq_one_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α Mul.mul LE.le\na : α\n⊢ a⁻ = 1 ↔ 1 ≤ a",
"tactic": "rw [le_antisymm_iff, neg_le_one_iff, inv_le_one', and_iff_left (one_le_neg _)]"
}
] |
[
239,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Algebra/Field/Power.lean
|
Odd.neg_one_zpow
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DivisionRing α\nn : ℤ\nh : Odd n\n⊢ (-1) ^ n = -1",
"tactic": "rw [h.neg_zpow, one_zpow]"
}
] |
[
41,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Embedding.map_constants
|
[] |
[
627,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
626,
1
] |
Std/Data/Option/Lemmas.lean
|
Option.liftOrGet_none_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → α → α\nb : Option α\n⊢ liftOrGet f none b = b",
"tactic": "cases b <;> rfl"
}
] |
[
185,
18
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
184,
9
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.mex_monotone
|
[
{
"state_after": "α✝ : Type ?u.377983\nβ✝ : Type ?u.377986\nγ : Type ?u.377989\nr : α✝ → α✝ → Prop\ns : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u\nf : α → Ordinal\ng : β → Ordinal\nh : range f ⊆ range g\ni : α\nhi : f i = mex g\n⊢ False",
"state_before": "α✝ : Type ?u.377983\nβ✝ : Type ?u.377986\nγ : Type ?u.377989\nr : α✝ → α✝ → Prop\ns : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u\nf : α → Ordinal\ng : β → Ordinal\nh : range f ⊆ range g\n⊢ mex f ≤ mex g",
"tactic": "refine' mex_le_of_ne fun i hi => _"
},
{
"state_after": "case intro\nα✝ : Type ?u.377983\nβ✝ : Type ?u.377986\nγ : Type ?u.377989\nr : α✝ → α✝ → Prop\ns : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u\nf : α → Ordinal\ng : β → Ordinal\nh : range f ⊆ range g\ni : α\nhi : f i = mex g\nj : β\nhj : g j = f i\n⊢ False",
"state_before": "α✝ : Type ?u.377983\nβ✝ : Type ?u.377986\nγ : Type ?u.377989\nr : α✝ → α✝ → Prop\ns : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u\nf : α → Ordinal\ng : β → Ordinal\nh : range f ⊆ range g\ni : α\nhi : f i = mex g\n⊢ False",
"tactic": "cases' h ⟨i, rfl⟩ with j hj"
},
{
"state_after": "case intro\nα✝ : Type ?u.377983\nβ✝ : Type ?u.377986\nγ : Type ?u.377989\nr : α✝ → α✝ → Prop\ns : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u\nf : α → Ordinal\ng : β → Ordinal\nh : range f ⊆ range g\ni : α\nj : β\nhi : g j = mex g\nhj : g j = f i\n⊢ False",
"state_before": "case intro\nα✝ : Type ?u.377983\nβ✝ : Type ?u.377986\nγ : Type ?u.377989\nr : α✝ → α✝ → Prop\ns : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u\nf : α → Ordinal\ng : β → Ordinal\nh : range f ⊆ range g\ni : α\nhi : f i = mex g\nj : β\nhj : g j = f i\n⊢ False",
"tactic": "rw [← hj] at hi"
},
{
"state_after": "no goals",
"state_before": "case intro\nα✝ : Type ?u.377983\nβ✝ : Type ?u.377986\nγ : Type ?u.377989\nr : α✝ → α✝ → Prop\ns : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u\nf : α → Ordinal\ng : β → Ordinal\nh : range f ⊆ range g\ni : α\nj : β\nhi : g j = mex g\nhj : g j = f i\n⊢ False",
"tactic": "exact ne_mex g j hi"
}
] |
[
2043,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2038,
1
] |
Mathlib/Data/Multiset/Lattice.lean
|
Multiset.sup_coe
|
[] |
[
38,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/RingTheory/Congruence.lean
|
RingCon.coe_zero
|
[] |
[
217,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_gt_univ
|
[] |
[
165,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Algebra/Hom/Equiv/Units/Basic.lean
|
Units.mapEquiv_symm
|
[] |
[
53,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.hom_eval₂
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝² : Semiring R\np q r : R[X]\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : R →+* S\ng : S →+* T\nx : S\n⊢ ↑g (eval₂ f x p) = eval₂ (RingHom.comp g f) (↑g x) p",
"tactic": "rw [← eval₂_map, eval₂_at_apply, eval_map]"
}
] |
[
1008,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1007,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
cauchySeq_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\nu : ℕ → α\n⊢ CauchySeq u ↔ ∀ (V : Set (α × α)), V ∈ 𝓤 α → ∃ N, ∀ (k : ℕ), k ≥ N → ∀ (l : ℕ), l ≥ N → (u k, u l) ∈ V",
"tactic": "simp only [cauchySeq_iff', Filter.eventually_atTop_prod_self', mem_preimage, Prod_map]"
}
] |
[
234,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.fuzzy_of_equiv_of_fuzzy
|
[] |
[
981,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
980,
1
] |
Mathlib/FieldTheory/IsAlgClosed/Basic.lean
|
IsAlgClosed.algebra_map_surjective_of_is_integral'
|
[] |
[
164,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.subset_set_smul_iff
|
[] |
[
925,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
922,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.inv_mul
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nz : ℤ_[p]\nhz : ‖z‖ = 1\n⊢ inv z * z = 1",
"tactic": "rw [mul_comm, mul_inv hz]"
}
] |
[
443,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_diff
|
[] |
[
105,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.nfpFamily_le_apply
|
[
{
"state_after": "ι : Type u\nf : ι → Ordinal → Ordinal\ninst✝ : Nonempty ι\nH : ∀ (i : ι), IsNormal (f i)\na b : Ordinal\n⊢ (¬∃ i, nfpFamily f a ≤ f i b) ↔ ¬nfpFamily f a ≤ b",
"state_before": "ι : Type u\nf : ι → Ordinal → Ordinal\ninst✝ : Nonempty ι\nH : ∀ (i : ι), IsNormal (f i)\na b : Ordinal\n⊢ (∃ i, nfpFamily f a ≤ f i b) ↔ nfpFamily f a ≤ b",
"tactic": "rw [← not_iff_not]"
},
{
"state_after": "ι : Type u\nf : ι → Ordinal → Ordinal\ninst✝ : Nonempty ι\nH : ∀ (i : ι), IsNormal (f i)\na b : Ordinal\n⊢ (∀ (i : ι), f i b < nfpFamily f a) ↔ b < nfpFamily f a",
"state_before": "ι : Type u\nf : ι → Ordinal → Ordinal\ninst✝ : Nonempty ι\nH : ∀ (i : ι), IsNormal (f i)\na b : Ordinal\n⊢ (¬∃ i, nfpFamily f a ≤ f i b) ↔ ¬nfpFamily f a ≤ b",
"tactic": "push_neg"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nf : ι → Ordinal → Ordinal\ninst✝ : Nonempty ι\nH : ∀ (i : ι), IsNormal (f i)\na b : Ordinal\n⊢ (∀ (i : ι), f i b < nfpFamily f a) ↔ b < nfpFamily f a",
"tactic": "exact apply_lt_nfpFamily_iff H"
}
] |
[
109,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/Finset/Image.lean
|
Function.Semiconj.finset_map
|
[] |
[
148,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.arcsin_le_iff_le_sin'
|
[
{
"state_after": "case inl\nx y : ℝ\nhy : y ∈ Ico (-(π / 2)) (π / 2)\nhx₁ : x ≤ -1\n⊢ arcsin x ≤ y ↔ x ≤ sin y\n\ncase inr\nx y : ℝ\nhy : y ∈ Ico (-(π / 2)) (π / 2)\nhx₁ : -1 ≤ x\n⊢ arcsin x ≤ y ↔ x ≤ sin y",
"state_before": "x y : ℝ\nhy : y ∈ Ico (-(π / 2)) (π / 2)\n⊢ arcsin x ≤ y ↔ x ≤ sin y",
"tactic": "cases' le_total x (-1) with hx₁ hx₁"
},
{
"state_after": "case inr.inl\nx y : ℝ\nhy : y ∈ Ico (-(π / 2)) (π / 2)\nhx₁ : -1 ≤ x\nhx₂ : 1 < x\n⊢ arcsin x ≤ y ↔ x ≤ sin y\n\ncase inr.inr\nx y : ℝ\nhy : y ∈ Ico (-(π / 2)) (π / 2)\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ arcsin x ≤ y ↔ x ≤ sin y",
"state_before": "case inr\nx y : ℝ\nhy : y ∈ Ico (-(π / 2)) (π / 2)\nhx₁ : -1 ≤ x\n⊢ arcsin x ≤ y ↔ x ≤ sin y",
"tactic": "cases' lt_or_le 1 x with hx₂ hx₂"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nx y : ℝ\nhy : y ∈ Ico (-(π / 2)) (π / 2)\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ arcsin x ≤ y ↔ x ≤ sin y",
"tactic": "exact arcsin_le_iff_le_sin ⟨hx₁, hx₂⟩ (mem_Icc_of_Ico hy)"
},
{
"state_after": "no goals",
"state_before": "case inl\nx y : ℝ\nhy : y ∈ Ico (-(π / 2)) (π / 2)\nhx₁ : x ≤ -1\n⊢ arcsin x ≤ y ↔ x ≤ sin y",
"tactic": "simp [arcsin_of_le_neg_one hx₁, hy.1, hx₁.trans (neg_one_le_sin _)]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nx y : ℝ\nhy : y ∈ Ico (-(π / 2)) (π / 2)\nhx₁ : -1 ≤ x\nhx₂ : 1 < x\n⊢ arcsin x ≤ y ↔ x ≤ sin y",
"tactic": "simp [arcsin_of_one_le hx₂.le, hy.2.not_le, (sin_le_one y).trans_lt hx₂]"
}
] |
[
159,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.hasBasis_nhdsSet_cthickening
|
[] |
[
1294,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1291,
1
] |
Mathlib/Algebra/Category/Mon/FilteredColimits.lean
|
MonCat.FilteredColimits.colimitMulAux_eq_of_rel_right
|
[
{
"state_after": "case mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx y' : (j : J) × ↑(F.obj j)\nj₁ : J\ny : ↑(F.obj j₁)\nhyy' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := y } y'\n⊢ colimitMulAux F x { fst := j₁, snd := y } = colimitMulAux F x y'",
"state_before": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx y y' : (j : J) × ↑(F.obj j)\nhyy' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) y y'\n⊢ colimitMulAux F x y = colimitMulAux F x y'",
"tactic": "cases' y with j₁ y"
},
{
"state_after": "case mk.mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\ny' : (j : J) × ↑(F.obj j)\nj₁ : J\ny : ↑(F.obj j₁)\nhyy' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := y } y'\nj₂ : J\nx : ↑(F.obj j₂)\n⊢ colimitMulAux F { fst := j₂, snd := x } { fst := j₁, snd := y } = colimitMulAux F { fst := j₂, snd := x } y'",
"state_before": "case mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx y' : (j : J) × ↑(F.obj j)\nj₁ : J\ny : ↑(F.obj j₁)\nhyy' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := y } y'\n⊢ colimitMulAux F x { fst := j₁, snd := y } = colimitMulAux F x y'",
"tactic": "cases' x with j₂ x"
},
{
"state_after": "case mk.mk.mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nhyy' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := y } { fst := j₃, snd := y' }\n⊢ colimitMulAux F { fst := j₂, snd := x } { fst := j₁, snd := y } =\n colimitMulAux F { fst := j₂, snd := x } { fst := j₃, snd := y' }",
"state_before": "case mk.mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\ny' : (j : J) × ↑(F.obj j)\nj₁ : J\ny : ↑(F.obj j₁)\nhyy' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := y } y'\nj₂ : J\nx : ↑(F.obj j₂)\n⊢ colimitMulAux F { fst := j₂, snd := x } { fst := j₁, snd := y } = colimitMulAux F { fst := j₂, snd := x } y'",
"tactic": "cases' y' with j₃ y'"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : (F ⋙ forget MonCat).map f { fst := j₁, snd := y }.snd = (F ⋙ forget MonCat).map g { fst := j₃, snd := y' }.snd\n⊢ colimitMulAux F { fst := j₂, snd := x } { fst := j₁, snd := y } =\n colimitMulAux F { fst := j₂, snd := x } { fst := j₃, snd := y' }",
"state_before": "case mk.mk.mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nhyy' : Types.FilteredColimit.Rel (F ⋙ forget MonCat) { fst := j₁, snd := y } { fst := j₃, snd := y' }\n⊢ colimitMulAux F { fst := j₂, snd := x } { fst := j₁, snd := y } =\n colimitMulAux F { fst := j₂, snd := x } { fst := j₃, snd := y' }",
"tactic": "obtain ⟨l, f, g, hfg⟩ := hyy'"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\n⊢ colimitMulAux F { fst := j₂, snd := x } { fst := j₁, snd := y } =\n colimitMulAux F { fst := j₂, snd := x } { fst := j₃, snd := y' }",
"state_before": "case mk.mk.mk.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : (F ⋙ forget MonCat).map f { fst := j₁, snd := y }.snd = (F ⋙ forget MonCat).map g { fst := j₃, snd := y' }.snd\n⊢ colimitMulAux F { fst := j₂, snd := x } { fst := j₁, snd := y } =\n colimitMulAux F { fst := j₂, snd := x } { fst := j₃, snd := y' }",
"tactic": "simp at hfg"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ colimitMulAux F { fst := j₂, snd := x } { fst := j₁, snd := y } =\n colimitMulAux F { fst := j₂, snd := x } { fst := j₃, snd := y' }",
"state_before": "case mk.mk.mk.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\n⊢ colimitMulAux F { fst := j₂, snd := x } { fst := j₁, snd := y } =\n colimitMulAux F { fst := j₂, snd := x } { fst := j₃, snd := y' }",
"tactic": "obtain ⟨s, α, β, γ, h₁, h₂, h₃⟩ :=\n IsFiltered.tulip (IsFiltered.rightToMax j₂ j₁) (IsFiltered.leftToMax j₂ j₁)\n (IsFiltered.leftToMax j₂ j₃) (IsFiltered.rightToMax j₂ j₃) f g"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ∃ k f g,\n ↑(F.map f)\n { fst := IsFiltered.max { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst))\n { fst := j₂, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst))\n { fst := j₁, snd := y }.snd }.snd =\n ↑(F.map g)\n { fst := IsFiltered.max { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst))\n { fst := j₂, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst))\n { fst := j₃, snd := y' }.snd }.snd",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ colimitMulAux F { fst := j₂, snd := x } { fst := j₁, snd := y } =\n colimitMulAux F { fst := j₂, snd := x } { fst := j₃, snd := y' }",
"tactic": "apply M.mk_eq"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map α)\n { fst := IsFiltered.max { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst))\n { fst := j₂, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst))\n { fst := j₁, snd := y }.snd }.snd =\n ↑(F.map γ)\n { fst := IsFiltered.max { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst))\n { fst := j₂, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst))\n { fst := j₃, snd := y' }.snd }.snd",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ∃ k f g,\n ↑(F.map f)\n { fst := IsFiltered.max { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst))\n { fst := j₂, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst))\n { fst := j₁, snd := y }.snd }.snd =\n ↑(F.map g)\n { fst := IsFiltered.max { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst))\n { fst := j₂, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst))\n { fst := j₃, snd := y' }.snd }.snd",
"tactic": "use s, α, γ"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₂ j₁)) x * ↑(F.map (IsFiltered.rightToMax j₂ j₁)) y) =\n ↑(F.map γ) (↑(F.map (IsFiltered.leftToMax j₂ j₃)) x * ↑(F.map (IsFiltered.rightToMax j₂ j₃)) y')",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map α)\n { fst := IsFiltered.max { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst))\n { fst := j₂, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₂, snd := x }.fst { fst := j₁, snd := y }.fst))\n { fst := j₁, snd := y }.snd }.snd =\n ↑(F.map γ)\n { fst := IsFiltered.max { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst))\n { fst := j₂, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₂, snd := x }.fst { fst := j₃, snd := y' }.fst))\n { fst := j₃, snd := y' }.snd }.snd",
"tactic": "dsimp"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₂ j₁)) x) * ↑(F.map α) (↑(F.map (IsFiltered.rightToMax j₂ j₁)) y) =\n ↑(F.map γ) (↑(F.map (IsFiltered.leftToMax j₂ j₃)) x) * ↑(F.map γ) (↑(F.map (IsFiltered.rightToMax j₂ j₃)) y')",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₂ j₁)) x * ↑(F.map (IsFiltered.rightToMax j₂ j₁)) y) =\n ↑(F.map γ) (↑(F.map (IsFiltered.leftToMax j₂ j₃)) x * ↑(F.map (IsFiltered.rightToMax j₂ j₃)) y')",
"tactic": "simp_rw [MonoidHom.map_mul]"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map (IsFiltered.leftToMax j₂ j₁) ≫ F.map α) x * ↑(F.map (IsFiltered.rightToMax j₂ j₁) ≫ F.map α) y =\n ↑(F.map (IsFiltered.leftToMax j₂ j₃) ≫ F.map γ) x * ↑(F.map (IsFiltered.rightToMax j₂ j₃) ≫ F.map γ) y'",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₂ j₁)) x) * ↑(F.map α) (↑(F.map (IsFiltered.rightToMax j₂ j₁)) y) =\n ↑(F.map γ) (↑(F.map (IsFiltered.leftToMax j₂ j₃)) x) * ↑(F.map γ) (↑(F.map (IsFiltered.rightToMax j₂ j₃)) y')",
"tactic": "change (F.map _ ≫ F.map _) _ * (F.map _ ≫ F.map _) _ =\n (F.map _ ≫ F.map _) _ * (F.map _ ≫ F.map _) _"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map (IsFiltered.leftToMax j₂ j₃) ≫ F.map γ) x * ↑(F.map f ≫ F.map β) y =\n ↑(F.map (IsFiltered.leftToMax j₂ j₃) ≫ F.map γ) x * ↑(F.map g ≫ F.map β) y'",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map (IsFiltered.leftToMax j₂ j₁) ≫ F.map α) x * ↑(F.map (IsFiltered.rightToMax j₂ j₁) ≫ F.map α) y =\n ↑(F.map (IsFiltered.leftToMax j₂ j₃) ≫ F.map γ) x * ↑(F.map (IsFiltered.rightToMax j₂ j₃) ≫ F.map γ) y'",
"tactic": "simp_rw [← F.map_comp, h₁, h₂, h₃, F.map_comp]"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h.e_a\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map f ≫ F.map β) y = ↑(F.map g ≫ F.map β) y'",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map (IsFiltered.leftToMax j₂ j₃) ≫ F.map γ) x * ↑(F.map f ≫ F.map β) y =\n ↑(F.map (IsFiltered.leftToMax j₂ j₃) ≫ F.map γ) x * ↑(F.map g ≫ F.map β) y'",
"tactic": "congr 1"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h.e_a\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map β) (↑(F.map f) y) = ↑(F.map β) (↑(F.map g) y')",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h.e_a\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map f ≫ F.map β) y = ↑(F.map g ≫ F.map β) y'",
"tactic": "change F.map _ (F.map _ _) = F.map _ (F.map _ _)"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.h.e_a\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nj₁ : J\ny : ↑(F.obj j₁)\nj₂ : J\nx : ↑(F.obj j₂)\nj₃ : J\ny' : ↑(F.obj j₃)\nl : J\nf : { fst := j₁, snd := y }.fst ⟶ l\ng : { fst := j₃, snd := y' }.fst ⟶ l\nhfg : ↑(F.map f) y = ↑(F.map g) y'\ns : J\nα : IsFiltered.max j₂ j₁ ⟶ s\nβ : l ⟶ s\nγ : IsFiltered.max j₂ j₃ ⟶ s\nh₁ : IsFiltered.rightToMax j₂ j₁ ≫ α = f ≫ β\nh₂ : IsFiltered.leftToMax j₂ j₁ ≫ α = IsFiltered.leftToMax j₂ j₃ ≫ γ\nh₃ : IsFiltered.rightToMax j₂ j₃ ≫ γ = g ≫ β\n⊢ ↑(F.map β) (↑(F.map f) y) = ↑(F.map β) (↑(F.map g) y')",
"tactic": "rw [hfg]"
}
] |
[
180,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Topology/Basic.lean
|
tendsto_const_nhds
|
[] |
[
1037,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1036,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.subset_erase
|
[] |
[
1947,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1945,
1
] |
Mathlib/Order/Antichain.lean
|
IsAntichain.greatest_iff
|
[] |
[
241,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.zero_smul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁶ : CommSemiring R\nR' : Type ?u.210413\ninst✝¹⁵ : Monoid R'\nR'' : Type u_4\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.210431\nQ : Type ?u.210434\nS : Type ?u.210437\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nx : M ⊗[R] N\nthis : ∀ (r : R'') (m : M) (n : N), r • m ⊗ₜ[R] n = (r • m) ⊗ₜ[R] n\n⊢ 0 • 0 = 0",
"tactic": "rw [TensorProduct.smul_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁶ : CommSemiring R\nR' : Type ?u.210413\ninst✝¹⁵ : Monoid R'\nR'' : Type u_4\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.210431\nQ : Type ?u.210434\nS : Type ?u.210437\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nx : M ⊗[R] N\nthis : ∀ (r : R'') (m : M) (n : N), r • m ⊗ₜ[R] n = (r • m) ⊗ₜ[R] n\nm : M\nn : N\n⊢ 0 • m ⊗ₜ[R] n = 0",
"tactic": "rw [this, zero_smul, zero_tmul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁶ : CommSemiring R\nR' : Type ?u.210413\ninst✝¹⁵ : Monoid R'\nR'' : Type u_4\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.210431\nQ : Type ?u.210434\nS : Type ?u.210437\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nx✝ : M ⊗[R] N\nthis : ∀ (r : R'') (m : M) (n : N), r • m ⊗ₜ[R] n = (r • m) ⊗ₜ[R] n\nx y : M ⊗[R] N\nihx : 0 • x = 0\nihy : 0 • y = 0\n⊢ 0 • (x + y) = 0",
"tactic": "rw [TensorProduct.smul_add, ihx, ihy, add_zero]"
}
] |
[
256,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
11
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.isLocallyFraction_pred
|
[] |
[
163,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.compl_mem_ae_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.98230\nγ : Type ?u.98233\nδ : Type ?u.98236\nι : Type ?u.98239\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t s : Set α\n⊢ sᶜ ∈ Measure.ae μ ↔ ↑↑μ s = 0",
"tactic": "simp only [mem_ae_iff, compl_compl]"
}
] |
[
382,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.nodalWeight_eq_eval_nodal_erase_inv
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ns : Finset ι\nv : ι → F\ni : ι\nr : ι → F\nx : F\ninst✝ : DecidableEq ι\n⊢ nodalWeight s v i = (eval (v i) (nodal (Finset.erase s i) v))⁻¹",
"tactic": "rw [eval_nodal, nodalWeight, prod_inv_distrib]"
}
] |
[
565,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_mul_cos_add_sin_mul_I_sub
|
[
{
"state_after": "r : ℝ\nhr : 0 < r\nθ : ℝ\n⊢ ↑(- -⌊(-π + 2 * π - θ) / (2 * π)⌋) * (2 * π) = 2 * π * ↑⌊(π - θ) / (2 * π)⌋",
"state_before": "r : ℝ\nhr : 0 < r\nθ : ℝ\n⊢ arg (↑r * (cos ↑θ + sin ↑θ * I)) - θ = 2 * π * ↑⌊(π - θ) / (2 * π)⌋",
"tactic": "rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor,\n zsmul_eq_mul]"
},
{
"state_after": "no goals",
"state_before": "r : ℝ\nhr : 0 < r\nθ : ℝ\n⊢ ↑(- -⌊(-π + 2 * π - θ) / (2 * π)⌋) * (2 * π) = 2 * π * ↑⌊(π - θ) / (2 * π)⌋",
"tactic": "ring_nf"
}
] |
[
450,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/Algebra/GroupWithZero/Power.lean
|
zero_zpow_eq
|
[
{
"state_after": "case inl\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\nn : ℤ\nh : n = 0\n⊢ 0 ^ n = 1\n\ncase inr\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\nn : ℤ\nh : ¬n = 0\n⊢ 0 ^ n = 0",
"state_before": "G₀ : Type u_1\ninst✝ : GroupWithZero G₀\nn : ℤ\n⊢ 0 ^ n = if n = 0 then 1 else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\nn : ℤ\nh : n = 0\n⊢ 0 ^ n = 1",
"tactic": "rw [h, zpow_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr\nG₀ : Type u_1\ninst✝ : GroupWithZero G₀\nn : ℤ\nh : ¬n = 0\n⊢ 0 ^ n = 0",
"tactic": "rw [zero_zpow _ h]"
}
] |
[
74,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Order/Basic.lean
|
subrelation_iff_le
|
[] |
[
1402,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1401,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
|
Submodule.orthogonal_gc
|
[] |
[
151,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Algebra/CharZero/Defs.lean
|
Nat.cast_eq_one
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : AddMonoidWithOne R\ninst✝ : CharZero R\nn : ℕ\n⊢ ↑n = 1 ↔ n = 1",
"tactic": "rw [← cast_one, cast_inj]"
}
] |
[
92,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Algebra/BigOperators/Pi.lean
|
MonoidHom.functions_ext'
|
[] |
[
105,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.quot_mk_to_coe
|
[] |
[
45,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Data/Finset/Option.lean
|
Finset.eraseNone_empty
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.12714\na✝ : α\n⊢ a✝ ∈ ↑eraseNone ∅ ↔ a✝ ∈ ∅",
"state_before": "α : Type u_1\nβ : Type ?u.12714\n⊢ ↑eraseNone ∅ = ∅",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.12714\na✝ : α\n⊢ a✝ ∈ ↑eraseNone ∅ ↔ a✝ ∈ ∅",
"tactic": "simp"
}
] |
[
136,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/Matrix/Hadamard.lean
|
Matrix.hadamard_smul
|
[] |
[
95,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/CategoryTheory/SingleObj.lean
|
Units.toAut_inv
|
[] |
[
213,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.map_add
|
[] |
[
101,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Topology/PathConnected.lean
|
Joined.symm
|
[] |
[
784,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
783,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
WithSeminorms.tendsto_nhds'
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.232913\n𝕝 : Type ?u.232916\n𝕝₂ : Type ?u.232919\nE : Type u_2\nF : Type u_4\nG : Type ?u.232928\nι : Type u_3\nι' : Type ?u.232934\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nu : F → E\nf : Filter F\ny₀ : E\n⊢ Filter.Tendsto u f (𝓝 y₀) ↔ ∀ (s : Finset ι) (ε : ℝ), 0 < ε → ∀ᶠ (x : F) in f, ↑(Finset.sup s p) (u x - y₀) < ε",
"tactic": "simp [hp.hasBasis_ball.tendsto_right_iff]"
}
] |
[
386,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Data/Matrix/Notation.lean
|
Matrix.one_fin_two
|
[
{
"state_after": "case a.h\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : Zero α\ninst✝ : One α\ni j : Fin 2\n⊢ OfNat.ofNat 1 i j = ↑of ![![1, 0], ![0, 1]] i j",
"state_before": "α : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : Zero α\ninst✝ : One α\n⊢ 1 = ↑of ![![1, 0], ![0, 1]]",
"tactic": "ext (i j)"
},
{
"state_after": "no goals",
"state_before": "case a.h\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : Zero α\ninst✝ : One α\ni j : Fin 2\n⊢ OfNat.ofNat 1 i j = ↑of ![![1, 0], ![0, 1]] i j",
"tactic": "fin_cases i <;> fin_cases j <;> rfl"
}
] |
[
413,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factors_count_eq
|
[
{
"state_after": "case inl\np : ℕ\n⊢ count p (factors 0) = ↑(factorization 0) p\n\ncase inr\nn p : ℕ\nhn0 : n > 0\n⊢ count p (factors n) = ↑(factorization n) p",
"state_before": "n p : ℕ\n⊢ count p (factors n) = ↑(factorization n) p",
"tactic": "rcases n.eq_zero_or_pos with (rfl | hn0)"
},
{
"state_after": "case pos\nn p : ℕ\nhn0 : n > 0\npp : Prime p\n⊢ count p (factors n) = ↑(factorization n) p\n\ncase neg\nn p : ℕ\nhn0 : n > 0\npp : ¬Prime p\n⊢ count p (factors n) = ↑(factorization n) p",
"state_before": "case inr\nn p : ℕ\nhn0 : n > 0\n⊢ count p (factors n) = ↑(factorization n) p",
"tactic": "by_cases pp : p.Prime"
},
{
"state_after": "case pos\nn p : ℕ\nhn0 : n > 0\npp : Prime p\n⊢ count p (factors n) = ↑(factorization n) p",
"state_before": "case pos\nn p : ℕ\nhn0 : n > 0\npp : Prime p\n⊢ count p (factors n) = ↑(factorization n) p\n\ncase neg\nn p : ℕ\nhn0 : n > 0\npp : ¬Prime p\n⊢ count p (factors n) = ↑(factorization n) p",
"tactic": "case neg =>\n rw [count_eq_zero_of_not_mem (mt prime_of_mem_factors pp)]\n simp [factorization, pp]"
},
{
"state_after": "case pos\nn p : ℕ\nhn0 : n > 0\npp : Prime p\n⊢ count p (factors n) = padicValNat p n",
"state_before": "case pos\nn p : ℕ\nhn0 : n > 0\npp : Prime p\n⊢ count p (factors n) = ↑(factorization n) p",
"tactic": "simp only [factorization, coe_mk, pp, if_true]"
},
{
"state_after": "case pos\nn p : ℕ\nhn0 : n > 0\npp : Prime p\n⊢ ↑(count p (factors n)) = ↑(Multiset.count (↑normalize p) (UniqueFactorizationMonoid.normalizedFactors n))",
"state_before": "case pos\nn p : ℕ\nhn0 : n > 0\npp : Prime p\n⊢ count p (factors n) = padicValNat p n",
"tactic": "rw [← PartENat.natCast_inj, padicValNat_def' pp.ne_one hn0,\n UniqueFactorizationMonoid.multiplicity_eq_count_normalizedFactors pp hn0.ne']"
},
{
"state_after": "no goals",
"state_before": "case pos\nn p : ℕ\nhn0 : n > 0\npp : Prime p\n⊢ ↑(count p (factors n)) = ↑(Multiset.count (↑normalize p) (UniqueFactorizationMonoid.normalizedFactors n))",
"tactic": "simp [factors_eq]"
},
{
"state_after": "no goals",
"state_before": "case inl\np : ℕ\n⊢ count p (factors 0) = ↑(factorization 0) p",
"tactic": "simp [factorization, count]"
},
{
"state_after": "n p : ℕ\nhn0 : n > 0\npp : ¬Prime p\n⊢ 0 = ↑(factorization n) p",
"state_before": "n p : ℕ\nhn0 : n > 0\npp : ¬Prime p\n⊢ count p (factors n) = ↑(factorization n) p",
"tactic": "rw [count_eq_zero_of_not_mem (mt prime_of_mem_factors pp)]"
},
{
"state_after": "no goals",
"state_before": "n p : ℕ\nhn0 : n > 0\npp : ¬Prime p\n⊢ 0 = ↑(factorization n) p",
"tactic": "simp [factorization, pp]"
}
] |
[
86,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.induction₂_symm_neg
|
[] |
[
1023,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1014,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
WithSeminorms.mem_nhds_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.207750\n𝕝 : Type ?u.207753\n𝕝₂ : Type ?u.207756\nE : Type u_2\nF : Type ?u.207762\nG : Type ?u.207765\nι : Type u_3\nι' : Type ?u.207771\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Nonempty ι\ninst✝ : TopologicalSpace E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nx : E\nU : Set E\n⊢ U ∈ 𝓝 x ↔ ∃ s r, r > 0 ∧ ball (Finset.sup s p) x r ⊆ U",
"tactic": "rw [hp.hasBasis_ball.mem_iff, Prod.exists]"
}
] |
[
326,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.inv_mul_of_invertible
|
[] |
[
351,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.map_smulₛₗ
|
[] |
[
966,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
965,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.comap_radical
|
[
{
"state_after": "case h\nR : Type u\nS : Type v\nF : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nx✝ : R\n⊢ x✝ ∈ comap f (radical K) ↔ x✝ ∈ radical (comap f K)",
"state_before": "R : Type u\nS : Type v\nF : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\n⊢ comap f (radical K) = radical (comap f K)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nS : Type v\nF : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nx✝ : R\n⊢ x✝ ∈ comap f (radical K) ↔ x✝ ∈ radical (comap f K)",
"tactic": "simp [radical]"
}
] |
[
1808,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1806,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.half_pos
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.325183\nβ : Type ?u.325186\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nh : a ≠ 0\n⊢ 0 < a / 2",
"tactic": "simp [h]"
}
] |
[
1744,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1744,
11
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
OmegaCompletePartialOrder.ContinuousHom.ite_continuous'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nα' : Type ?u.55194\nβ : Type v\nβ' : Type ?u.55199\nγ : Type ?u.55202\nφ : Type ?u.55205\ninst✝⁵ : OmegaCompletePartialOrder α\ninst✝⁴ : OmegaCompletePartialOrder β\ninst✝³ : OmegaCompletePartialOrder γ\ninst✝² : OmegaCompletePartialOrder φ\ninst✝¹ : OmegaCompletePartialOrder α'\ninst✝ : OmegaCompletePartialOrder β'\np : Prop\nhp : Decidable p\nf g : α → β\nhf : Continuous' f\nhg : Continuous' g\n⊢ Continuous' fun x => if p then f x else g x",
"tactic": "split_ifs <;> simp [*]"
}
] |
[
624,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
622,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
rootsOfUnity_le_of_dvd
|
[
{
"state_after": "case intro\nM : Type u_1\nN : Type ?u.364389\nG : Type ?u.364392\nR : Type ?u.364395\nS : Type ?u.364398\nF : Type ?u.364401\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk d : ℕ+\n⊢ rootsOfUnity k M ≤ rootsOfUnity (k * d) M",
"state_before": "M : Type u_1\nN : Type ?u.364389\nG : Type ?u.364392\nR : Type ?u.364395\nS : Type ?u.364398\nF : Type ?u.364401\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ+\nh : k ∣ l\n⊢ rootsOfUnity k M ≤ rootsOfUnity l M",
"tactic": "obtain ⟨d, rfl⟩ := h"
},
{
"state_after": "case intro\nM : Type u_1\nN : Type ?u.364389\nG : Type ?u.364392\nR : Type ?u.364395\nS : Type ?u.364398\nF : Type ?u.364401\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk d : ℕ+\nζ : Mˣ\nh : ζ ∈ rootsOfUnity k M\n⊢ ζ ∈ rootsOfUnity (k * d) M",
"state_before": "case intro\nM : Type u_1\nN : Type ?u.364389\nG : Type ?u.364392\nR : Type ?u.364395\nS : Type ?u.364398\nF : Type ?u.364401\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk d : ℕ+\n⊢ rootsOfUnity k M ≤ rootsOfUnity (k * d) M",
"tactic": "intro ζ h"
},
{
"state_after": "no goals",
"state_before": "case intro\nM : Type u_1\nN : Type ?u.364389\nG : Type ?u.364392\nR : Type ?u.364395\nS : Type ?u.364398\nF : Type ?u.364401\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk d : ℕ+\nζ : Mˣ\nh : ζ ∈ rootsOfUnity k M\n⊢ ζ ∈ rootsOfUnity (k * d) M",
"tactic": "simp_all only [mem_rootsOfUnity, PNat.mul_coe, pow_mul, one_pow]"
}
] |
[
125,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_singleton_ne_top
|
[] |
[
202,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Computability/Reduce.lean
|
Computable.equiv₂
|
[] |
[
230,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
LinearMap.ofIsCompl_left_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁹ : Ring R\nE : Type u_1\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.199427\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.200390\ninst✝² : Semiring S\nM : Type ?u.200396\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nh : IsCompl p q\nφ : { x // x ∈ p } →ₗ[R] F\nψ : { x // x ∈ q } →ₗ[R] F\nu : { x // x ∈ p }\n⊢ ↑(ofIsCompl h φ ψ) ↑u = ↑φ u",
"tactic": "simp [ofIsCompl]"
}
] |
[
237,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3963\ns t : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\n⊢ ncard s = 0 ↔ s = ∅",
"tactic": "rw [ncard_def, @Finite.card_eq_zero_iff _ hs.to_subtype, isEmpty_subtype,\n eq_empty_iff_forall_not_mem]"
}
] |
[
89,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
9
] |
Mathlib/Topology/SubsetProperties.lean
|
IsClopen.inter
|
[] |
[
1558,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1557,
1
] |
Std/Logic.lean
|
Decidable.or_iff_not_imp_left
|
[] |
[
544,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
543,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Comp.lean
|
HasFDerivWithinAt.comp_of_mem
|
[] |
[
109,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
add_le_of_le_tsub_right_of_le
|
[] |
[
39,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.compl_mem_cofinite
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.546970\nγ : Type ?u.546973\nδ : Type ?u.546976\nι : Type ?u.546979\nR : Type ?u.546982\nR' : Type ?u.546985\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\n⊢ sᶜ ∈ cofinite μ ↔ ↑↑μ s < ⊤",
"tactic": "rw [mem_cofinite, compl_compl]"
}
] |
[
2641,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2641,
1
] |
Mathlib/Analysis/SpecialFunctions/Sqrt.lean
|
Real.hasDerivAt_sqrt
|
[] |
[
72,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Algebra/FreeMonoid/Count.lean
|
FreeAddMonoid.count_apply
|
[] |
[
53,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Analysis/Complex/Schwarz.lean
|
Complex.dist_le_dist_of_mapsTo_ball_self
|
[
{
"state_after": "f : ℂ → ℂ\nc z : ℂ\nR R₁ R₂ : ℝ\nhd : DifferentiableOn ℂ f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nhz : z ∈ ball c R\nthis : dist (f z) (f c) ≤ R / R * dist z c\n⊢ dist (f z) c ≤ dist z c",
"state_before": "f : ℂ → ℂ\nc z : ℂ\nR R₁ R₂ : ℝ\nhd : DifferentiableOn ℂ f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nhz : z ∈ ball c R\n⊢ dist (f z) c ≤ dist z c",
"tactic": "have := dist_le_div_mul_dist_of_mapsTo_ball hd (by rwa [hc]) hz"
},
{
"state_after": "f : ℂ → ℂ\nc z : ℂ\nR R₁ R₂ : ℝ\nhd : DifferentiableOn ℂ f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nhz : z ∈ ball c R\nthis : dist (f z) c ≤ R / R * dist z c\n⊢ R ≠ 0",
"state_before": "f : ℂ → ℂ\nc z : ℂ\nR R₁ R₂ : ℝ\nhd : DifferentiableOn ℂ f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nhz : z ∈ ball c R\nthis : dist (f z) (f c) ≤ R / R * dist z c\n⊢ dist (f z) c ≤ dist z c",
"tactic": "rwa [hc, div_self, one_mul] at this"
},
{
"state_after": "no goals",
"state_before": "f : ℂ → ℂ\nc z : ℂ\nR R₁ R₂ : ℝ\nhd : DifferentiableOn ℂ f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nhz : z ∈ ball c R\nthis : dist (f z) c ≤ R / R * dist z c\n⊢ R ≠ 0",
"tactic": "exact (nonempty_ball.1 ⟨z, hz⟩).ne'"
},
{
"state_after": "no goals",
"state_before": "f : ℂ → ℂ\nc z : ℂ\nR R₁ R₂ : ℝ\nhd : DifferentiableOn ℂ f (ball c R)\nh_maps : MapsTo f (ball c R) (ball c R)\nhc : f c = c\nhz : z ∈ ball c R\n⊢ MapsTo f (ball c R) (ball (f c) ?m.43512)",
"tactic": "rwa [hc]"
}
] |
[
191,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.support_mul
|
[
{
"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✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\ninst✝ : DecidableEq σ\np q : MvPolynomial σ R\n⊢ support (p * q) ⊆ Finset.biUnion (support p) fun a => Finset.biUnion (support q) fun b => {a + b}",
"tactic": "convert AddMonoidAlgebra.support_mul p q"
}
] |
[
597,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
595,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_eq_iSup_nat
|
[] |
[
848,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
846,
11
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.IsComplement'.disjoint
|
[] |
[
511,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean
|
Matrix.det_transvection_of_ne
|
[
{
"state_after": "no goals",
"state_before": "n : Type u_1\np : Type ?u.20141\nR : Type u₂\n𝕜 : Type ?u.20146\ninst✝⁴ : Field 𝕜\ninst✝³ : DecidableEq n\ninst✝² : DecidableEq p\ninst✝¹ : CommRing R\ni j : n\ninst✝ : Fintype n\nh : i ≠ j\nc : R\n⊢ det (transvection i j c) = 1",
"tactic": "rw [← updateRow_eq_transvection i j, det_updateRow_add_smul_self _ h, det_one]"
}
] |
[
146,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
Finsupp.prod_comm
|
[] |
[
103,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.coeff_inj
|
[] |
[
665,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.subtypeEquiv_refl
|
[
{
"state_after": "case H.a\nα : Sort u_1\np : α → Prop\nh : optParam (∀ (a : α), p a ↔ p (↑(Equiv.refl α) a)) (_ : ∀ (a : α), p a ↔ p a)\nx✝ : { a // p a }\n⊢ ↑(↑(subtypeEquiv (Equiv.refl α) h) x✝) = ↑(↑(Equiv.refl { a // p a }) x✝)",
"state_before": "α : Sort u_1\np : α → Prop\nh : optParam (∀ (a : α), p a ↔ p (↑(Equiv.refl α) a)) (_ : ∀ (a : α), p a ↔ p a)\n⊢ subtypeEquiv (Equiv.refl α) h = Equiv.refl { a // p a }",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H.a\nα : Sort u_1\np : α → Prop\nh : optParam (∀ (a : α), p a ↔ p (↑(Equiv.refl α) a)) (_ : ∀ (a : α), p a ↔ p a)\nx✝ : { a // p a }\n⊢ ↑(↑(subtypeEquiv (Equiv.refl α) h) x✝) = ↑(↑(Equiv.refl { a // p a }) x✝)",
"tactic": "rfl"
}
] |
[
1090,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1087,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.offDiag_subset_prod
|
[] |
[
566,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
566,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.lift_iSup_le_lift_iSup'
|
[] |
[
1225,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1222,
1
] |
Mathlib/Topology/Covering.lean
|
IsCoveringMap.mk
|
[] |
[
159,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
exists_nhds_split_inv
|
[
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : TopologicalSpace α\nf : α → G\ns✝ : Set α\nx : α\ns : Set G\nhs : s ∈ 𝓝 1\nthis : (fun p => p.fst * p.snd⁻¹) ⁻¹' s ∈ 𝓝 (1, 1)\n⊢ ∃ V, V ∈ 𝓝 1 ∧ ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v / w ∈ s",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : TopologicalSpace α\nf : α → G\ns✝ : Set α\nx : α\ns : Set G\nhs : s ∈ 𝓝 1\n⊢ ∃ V, V ∈ 𝓝 1 ∧ ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v / w ∈ s",
"tactic": "have : (fun p : G × G => p.1 * p.2⁻¹) ⁻¹' s ∈ 𝓝 ((1, 1) : G × G) :=\n continuousAt_fst.mul continuousAt_snd.inv (by simpa)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : TopologicalSpace α\nf : α → G\ns✝ : Set α\nx : α\ns : Set G\nhs : s ∈ 𝓝 1\nthis : (fun p => p.fst * p.snd⁻¹) ⁻¹' s ∈ 𝓝 (1, 1)\n⊢ ∃ V, V ∈ 𝓝 1 ∧ ∀ (v : G), v ∈ V → ∀ (w : G), w ∈ V → v / w ∈ s",
"tactic": "simpa only [div_eq_mul_inv, nhds_prod_eq, mem_prod_self_iff, prod_subset_iff, mem_preimage] using\n this"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : TopologicalGroup G\ninst✝ : TopologicalSpace α\nf : α → G\ns✝ : Set α\nx : α\ns : Set G\nhs : s ∈ 𝓝 1\n⊢ s ∈ 𝓝 ((fun x => x.fst * x.snd⁻¹) (1, 1))",
"tactic": "simpa"
}
] |
[
809,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
804,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.subtypeDomain_sub
|
[] |
[
1142,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1141,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.get_eq_iff_eq_some
|
[] |
[
205,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
8
] |
Mathlib/Topology/MetricSpace/IsometricSMul.lean
|
edist_smul_left
|
[] |
[
82,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/RingTheory/Valuation/Integers.lean
|
Valuation.Integers.dvd_iff_le
|
[] |
[
130,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.subset_set_biUnion_of_mem
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.467741\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.467750\nι : Type ?u.467753\nκ : Type ?u.467756\ns : Finset α\nf : α → Set β\nx : α\nh : x ∈ s\n⊢ f x ≤ ⨆ (_ : x ∈ s), f x",
"tactic": "simp only [h, iSup_pos, le_refl]"
}
] |
[
2032,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2030,
1
] |
Mathlib/Deprecated/Group.lean
|
IsGroupHom.id
|
[] |
[
315,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Data/Bitvec/Lemmas.lean
|
Bitvec.toNat_ofNat
|
[
{
"state_after": "case zero\nn✝ n : ℕ\n⊢ Bitvec.toNat (Bitvec.ofNat zero n) = n % 2 ^ zero\n\ncase succ\nn✝ k : ℕ\nih : ∀ {n : ℕ}, Bitvec.toNat (Bitvec.ofNat k n) = n % 2 ^ k\nn : ℕ\n⊢ Bitvec.toNat (Bitvec.ofNat (succ k) n) = n % 2 ^ succ k",
"state_before": "k n : ℕ\n⊢ Bitvec.toNat (Bitvec.ofNat k n) = n % 2 ^ k",
"tactic": "induction' k with k ih generalizing n"
},
{
"state_after": "case zero\nn✝ n : ℕ\n⊢ Bitvec.toNat (Bitvec.ofNat 0 n) = 0",
"state_before": "case zero\nn✝ n : ℕ\n⊢ Bitvec.toNat (Bitvec.ofNat zero n) = n % 2 ^ zero",
"tactic": "simp [Nat.mod_one]"
},
{
"state_after": "no goals",
"state_before": "case zero\nn✝ n : ℕ\n⊢ Bitvec.toNat (Bitvec.ofNat 0 n) = 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case succ\nn✝ k : ℕ\nih : ∀ {n : ℕ}, Bitvec.toNat (Bitvec.ofNat k n) = n % 2 ^ k\nn : ℕ\n⊢ Bitvec.toNat (Bitvec.ofNat (succ k) n) = n % 2 ^ succ k",
"tactic": "rw [ofNat_succ, toNat_append, ih, bits_toNat_decide, mod_pow_succ, Nat.mul_comm]"
}
] |
[
77,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis
|
[
{
"state_after": "case mp\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\n⊢ IsCompact U ∧ IsOpen U → ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i\n\ncase mpr\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\n⊢ (∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i) → IsCompact U ∧ IsOpen U",
"state_before": "α : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\n⊢ IsCompact U ∧ IsOpen U ↔ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"state_before": "case mp\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\n⊢ IsCompact U ∧ IsOpen U → ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"tactic": "rintro ⟨h₁, h₂⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf : β → Set α\ne : U = ⋃ (i : β), f i\nhf : ∀ (i : β), f i ∈ range b\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"state_before": "case mp.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"tactic": "obtain ⟨β, f, e, hf⟩ := hb.open_eq_iUnion h₂"
},
{
"state_after": "case mp.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf : β → Set α\ne : U = ⋃ (i : β), f i\nf' : β → ι\nhf' : ∀ (i : β), b (f' i) = f i\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"state_before": "case mp.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf : β → Set α\ne : U = ⋃ (i : β), f i\nhf : ∀ (i : β), f i ∈ range b\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"tactic": "choose f' hf' using hf"
},
{
"state_after": "case mp.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf : β → Set α\ne : U = ⋃ (i : β), f i\nf' : β → ι\nhf' : ∀ (i : β), b (f' i) = f i\nthis : b ∘ f' = f\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"state_before": "case mp.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf : β → Set α\ne : U = ⋃ (i : β), f i\nf' : β → ι\nhf' : ∀ (i : β), b (f' i) = f i\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"tactic": "have : b ∘ f' = f := funext hf'"
},
{
"state_after": "case mp.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"state_before": "case mp.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf : β → Set α\ne : U = ⋃ (i : β), f i\nf' : β → ι\nhf' : ∀ (i : β), b (f' i) = f i\nthis : b ∘ f' = f\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"tactic": "subst this"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"state_before": "case mp.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"tactic": "obtain ⟨t, ht⟩ :=\n h₁.elim_finite_subcover (b ∘ f') (fun i => hb.isOpen (Set.mem_range_self _)) (by rw [e])"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_1\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ U ≤ ⋃ (i : ι) (_ : i ∈ ↑(Finset.image f' t)), b i\n\ncase mp.intro.intro.intro.intro.intro.refine'_2\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ (⋃ (i : ι) (_ : i ∈ ↑(Finset.image f' t)), b i) ≤ U",
"state_before": "case mp.intro.intro.intro.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ ∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i",
"tactic": "refine' ⟨t.image f', Set.Finite.intro inferInstance, le_antisymm _ _⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\n⊢ U ⊆ ⋃ (i : β), (b ∘ f') i",
"tactic": "rw [e]"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_1\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ (⋃ (i : β) (_ : i ∈ t), (b ∘ f') i) ⊆ ⋃ (i : ι) (_ : i ∈ ↑(Finset.image f' t)), b i",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_1\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ U ≤ ⋃ (i : ι) (_ : i ∈ ↑(Finset.image f' t)), b i",
"tactic": "refine' Set.Subset.trans ht _"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_1\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ ∀ (i : β), i ∈ t → (b ∘ f') i ⊆ ⋃ (i : ι) (_ : i ∈ ↑(Finset.image f' t)), b i",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_1\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ (⋃ (i : β) (_ : i ∈ t), (b ∘ f') i) ⊆ ⋃ (i : ι) (_ : i ∈ ↑(Finset.image f' t)), b i",
"tactic": "simp only [Set.iUnion_subset_iff]"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_1\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\ni : β\nhi : i ∈ t\n⊢ (b ∘ f') i ⊆ ⋃ (i : ι) (_ : i ∈ ↑(Finset.image f' t)), b i",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_1\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ ∀ (i : β), i ∈ t → (b ∘ f') i ⊆ ⋃ (i : ι) (_ : i ∈ ↑(Finset.image f' t)), b i",
"tactic": "intro i hi"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_1\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\ni : β\nhi : i ∈ t\n⊢ (b ∘ f') i ⊆ ⋃ (x : { x // x ∈ Finset.image f' t }), b ↑x",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_1\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\ni : β\nhi : i ∈ t\n⊢ (b ∘ f') i ⊆ ⋃ (i : ι) (_ : i ∈ ↑(Finset.image f' t)), b i",
"tactic": "erw [← Set.iUnion_subtype (fun x : ι => x ∈ t.image f') fun i => b i.1]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_1\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\ni : β\nhi : i ∈ t\n⊢ (b ∘ f') i ⊆ ⋃ (x : { x // x ∈ Finset.image f' t }), b ↑x",
"tactic": "exact Set.subset_iUnion (fun i : t.image f' => b i) ⟨_, Finset.mem_image_of_mem _ hi⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_2.h\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ ∀ (i : ι), i ∈ ↑(Finset.image f' t) → b i ⊆ U",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ (⋃ (i : ι) (_ : i ∈ ↑(Finset.image f' t)), b i) ≤ U",
"tactic": "apply Set.iUnion₂_subset"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_2.h\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\ni : ι\nhi : i ∈ ↑(Finset.image f' t)\n⊢ b i ⊆ U",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2.h\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\n⊢ ∀ (i : ι), i ∈ ↑(Finset.image f' t) → b i ⊆ U",
"tactic": "rintro i hi"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_2.h.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\nj : β\nhi : f' j ∈ ↑(Finset.image f' t)\n⊢ b (f' j) ⊆ U",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2.h\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\ni : ι\nhi : i ∈ ↑(Finset.image f' t)\n⊢ b i ⊆ U",
"tactic": "obtain ⟨j, -, rfl⟩ := Finset.mem_image.mp hi"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.refine'_2.h.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\nj : β\nhi : f' j ∈ ↑(Finset.image f' t)\n⊢ b (f' j) ⊆ ⋃ (i : β), (b ∘ f') i",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2.h.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\nj : β\nhi : f' j ∈ ↑(Finset.image f' t)\n⊢ b (f' j) ⊆ U",
"tactic": "rw [e]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro.refine'_2.h.intro.intro\nα : Type u\nβ✝ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\ns t✝ : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\nh₁ : IsCompact U\nh₂ : IsOpen U\nβ : Type u\nf' : β → ι\ne : U = ⋃ (i : β), (b ∘ f') i\nhf' : ∀ (i : β), b (f' i) = (b ∘ f') i\nt : Finset β\nht : U ⊆ ⋃ (i : β) (_ : i ∈ t), (b ∘ f') i\nj : β\nhi : f' j ∈ ↑(Finset.image f' t)\n⊢ b (f' j) ⊆ ⋃ (i : β), (b ∘ f') i",
"tactic": "exact Set.subset_iUnion (b ∘ f') j"
},
{
"state_after": "case mpr.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\ns : Set ι\nhs : Set.Finite s\n⊢ IsCompact (⋃ (i : ι) (_ : i ∈ s), b i) ∧ IsOpen (⋃ (i : ι) (_ : i ∈ s), b i)",
"state_before": "case mpr\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\nU : Set α\n⊢ (∃ s, Set.Finite s ∧ U = ⋃ (i : ι) (_ : i ∈ s), b i) → IsCompact U ∧ IsOpen U",
"tactic": "rintro ⟨s, hs, rfl⟩"
},
{
"state_after": "case mpr.intro.intro.left\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\ns : Set ι\nhs : Set.Finite s\n⊢ IsCompact (⋃ (i : ι) (_ : i ∈ s), b i)\n\ncase mpr.intro.intro.right\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\ns : Set ι\nhs : Set.Finite s\n⊢ IsOpen (⋃ (i : ι) (_ : i ∈ s), b i)",
"state_before": "case mpr.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\ns : Set ι\nhs : Set.Finite s\n⊢ IsCompact (⋃ (i : ι) (_ : i ∈ s), b i) ∧ IsOpen (⋃ (i : ι) (_ : i ∈ s), b i)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.left\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\ns : Set ι\nhs : Set.Finite s\n⊢ IsCompact (⋃ (i : ι) (_ : i ∈ s), b i)",
"tactic": "exact hs.isCompact_biUnion fun i _ => hb' i"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.right\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type ?u.42425\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t : Set α\nb : ι → Set α\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsCompact (b i)\ns : Set ι\nhs : Set.Finite s\n⊢ IsOpen (⋃ (i : ι) (_ : i ∈ s), b i)",
"tactic": "exact isOpen_biUnion fun i _ => hb.isOpen (Set.mem_range_self _)"
}
] |
[
511,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.map_monotone
|
[
{
"state_after": "ι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\n⊢ Subgraph.map f H ≤ Subgraph.map f H'",
"state_before": "ι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\n⊢ Monotone (Subgraph.map f)",
"tactic": "intro H H' h"
},
{
"state_after": "case left\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\n⊢ (Subgraph.map f H).verts ⊆ (Subgraph.map f H').verts\n\ncase right\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\n⊢ ∀ ⦃v w : W⦄, Adj (Subgraph.map f H) v w → Adj (Subgraph.map f H') v w",
"state_before": "ι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\n⊢ Subgraph.map f H ≤ Subgraph.map f H'",
"tactic": "constructor"
},
{
"state_after": "case left\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\na✝ : W\n⊢ a✝ ∈ (Subgraph.map f H).verts → a✝ ∈ (Subgraph.map f H').verts",
"state_before": "case left\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\n⊢ (Subgraph.map f H).verts ⊆ (Subgraph.map f H').verts",
"tactic": "intro"
},
{
"state_after": "case left\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\na✝ : W\n⊢ ∀ (x : V), x ∈ H.verts → ↑f x = a✝ → ∃ x, x ∈ H'.verts ∧ ↑f x = a✝",
"state_before": "case left\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\na✝ : W\n⊢ a✝ ∈ (Subgraph.map f H).verts → a✝ ∈ (Subgraph.map f H').verts",
"tactic": "simp only [map_verts, Set.mem_image, forall_exists_index, and_imp]"
},
{
"state_after": "case left\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\nv : V\nhv : v ∈ H.verts\n⊢ ∃ x, x ∈ H'.verts ∧ ↑f x = ↑f v",
"state_before": "case left\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\na✝ : W\n⊢ ∀ (x : V), x ∈ H.verts → ↑f x = a✝ → ∃ x, x ∈ H'.verts ∧ ↑f x = a✝",
"tactic": "rintro v hv rfl"
},
{
"state_after": "no goals",
"state_before": "case left\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\nv : V\nhv : v ∈ H.verts\n⊢ ∃ x, x ∈ H'.verts ∧ ↑f x = ↑f v",
"tactic": "exact ⟨_, h.1 hv, rfl⟩"
},
{
"state_after": "case right.intro.intro.intro.intro\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\nu v : V\nha : Adj H u v\n⊢ Adj (Subgraph.map f H') (↑f u) (↑f v)",
"state_before": "case right\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\n⊢ ∀ ⦃v w : W⦄, Adj (Subgraph.map f H) v w → Adj (Subgraph.map f H') v w",
"tactic": "rintro _ _ ⟨u, v, ha, rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case right.intro.intro.intro.intro\nι : Sort ?u.145982\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nG' : SimpleGraph W\nf : G →g G'\nH H' : Subgraph G\nh : H ≤ H'\nu v : V\nha : Adj H u v\n⊢ Adj (Subgraph.map f H') (↑f u) (↑f v)",
"tactic": "exact ⟨_, _, h.2 ha, rfl, rfl⟩"
}
] |
[
666,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/RingTheory/Ideal/Quotient.lean
|
Ideal.quotientInfEquivQuotientProd_snd
|
[] |
[
526,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.sameCycle_apply_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.73203\nα : Type u_1\nβ : Type ?u.73209\nf g : Perm α\np : α → Prop\nx y z : α\n⊢ (∃ b, ↑(f ^ ↑(Equiv.addRight 1).symm b) (↑f x) = y) ↔ SameCycle f x y",
"tactic": "simp [zpow_sub, SameCycle, Int.add_neg_one, Function.comp]"
}
] |
[
145,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Topology/PathConnected.lean
|
Path.continuous_extend
|
[] |
[
249,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_eq_bot_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.277101\nι : Sort x\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\n⊢ ∅ ∈ map m f → ∅ ∈ f",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.277101\nι : Sort x\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\n⊢ map m f = ⊥ → f = ⊥",
"tactic": "rw [← empty_mem_iff_bot, ← empty_mem_iff_bot]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.277101\nι : Sort x\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\n⊢ ∅ ∈ map m f → ∅ ∈ f",
"tactic": "exact id"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.277101\nι : Sort x\nf f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\nh : f = ⊥\n⊢ map m f = ⊥",
"tactic": "simp only [h, map_bot]"
}
] |
[
2449,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2446,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
contDiffWithinAt_univ
|
[] |
[
1323,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1322,
1
] |
Mathlib/Algebra/Free.lean
|
FreeSemigroup.traverse_mul
|
[
{
"state_after": "α β : Type u\nm : Type u → Type u\ninst✝¹ : Applicative m\nF : α → m β\ninst✝ : LawfulApplicative m\nx✝¹ y✝ : FreeSemigroup α\nx✝ : α\nL1 : List α\ny : α\nL2 : List α\nhd : α\ntl : List α\nih :\n ∀ (x : α),\n traverse F ({ head := x, tail := tl } * { head := y, tail := L2 }) =\n Seq.seq ((fun x x_1 => x * x_1) <$> traverse F { head := x, tail := tl }) fun x =>\n traverse F { head := y, tail := L2 }\nx : α\n⊢ (Seq.seq ((fun x x_1 => x * x_1) <$> pure <$> F x) fun x =>\n Seq.seq ((fun x x_1 => x * x_1) <$> traverse F { head := hd, tail := tl }) fun x =>\n traverse F { head := y, tail := L2 }) =\n Seq.seq\n ((fun x x_1 => x * x_1) <$>\n Seq.seq ((fun x x_1 => x * x_1) <$> pure <$> F x) fun x => traverse F { head := hd, tail := tl })\n fun x => traverse F { head := y, tail := L2 }",
"state_before": "α β : Type u\nm : Type u → Type u\ninst✝¹ : Applicative m\nF : α → m β\ninst✝ : LawfulApplicative m\nx✝¹ y✝ : FreeSemigroup α\nx✝ : α\nL1 : List α\ny : α\nL2 : List α\nhd : α\ntl : List α\nih :\n ∀ (x : α),\n traverse F ({ head := x, tail := tl } * { head := y, tail := L2 }) =\n Seq.seq ((fun x x_1 => x * x_1) <$> traverse F { head := x, tail := tl }) fun x =>\n traverse F { head := y, tail := L2 }\nx : α\n⊢ (Seq.seq ((fun x x_1 => x * x_1) <$> pure <$> F x) fun x =>\n traverse F ({ head := hd, tail := tl } * { head := y, tail := L2 })) =\n Seq.seq\n ((fun x x_1 => x * x_1) <$>\n Seq.seq ((fun x x_1 => x * x_1) <$> pure <$> F x) fun x => traverse F { head := hd, tail := tl })\n fun x => traverse F { head := y, tail := L2 }",
"tactic": "rw [ih]"
}
] |
[
666,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
657,
1
] |
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
|
MvPolynomial.IsHomogeneous.sum
|
[] |
[
187,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Data/Fin/Tuple/Sort.lean
|
Tuple.graphEquiv₂_apply
|
[] |
[
95,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.WalkingPair.equivBool_symm_apply_false
|
[] |
[
112,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Topology/Separation.lean
|
regularSpace_iInf
|
[] |
[
1622,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1620,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
Finset.aestronglyMeasurable_prod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.330452\nγ : Type ?u.330455\nι✝ : Type ?u.330458\ninst✝⁵ : Countable ι✝\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\nf✝ g : α → β\nM : Type u_3\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nι : Type u_1\nf : ι → α → M\ns : Finset ι\nhf : ∀ (i : ι), i ∈ s → AEStronglyMeasurable (f i) μ\n⊢ AEStronglyMeasurable (fun a => ∏ i in s, f i a) μ",
"tactic": "simpa only [← Finset.prod_apply] using s.aestronglyMeasurable_prod' hf"
}
] |
[
1428,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1425,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.coeff_zero
|
[] |
[
622,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean
|
MeasureTheory.withDensityᵥ_add
|
[
{
"state_after": "case h\nα : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(withDensityᵥ μ (f + g)) i = ↑(withDensityᵥ μ f + withDensityᵥ μ g) i",
"state_before": "α : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\n⊢ withDensityᵥ μ (f + g) = withDensityᵥ μ f + withDensityᵥ μ g",
"tactic": "ext1 i hi"
},
{
"state_after": "case h\nα : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\ni : Set α\nhi : MeasurableSet i\n⊢ (∫ (x : α) in i, (f + g) x ∂μ) = (∫ (x : α) in i, f x ∂μ) + ∫ (x : α) in i, g x ∂μ",
"state_before": "case h\nα : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(withDensityᵥ μ (f + g)) i = ↑(withDensityᵥ μ f + withDensityᵥ μ g) i",
"tactic": "rw [withDensityᵥ_apply (hf.add hg) hi, VectorMeasure.add_apply, withDensityᵥ_apply hf hi,\n withDensityᵥ_apply hg hi]"
},
{
"state_after": "case h\nα : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\ni : Set α\nhi : MeasurableSet i\n⊢ (∫ (x : α) in i, f x + g x ∂μ) = (∫ (x : α) in i, f x ∂μ) + ∫ (x : α) in i, g x ∂μ",
"state_before": "case h\nα : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\ni : Set α\nhi : MeasurableSet i\n⊢ (∫ (x : α) in i, (f + g) x ∂μ) = (∫ (x : α) in i, f x ∂μ) + ∫ (x : α) in i, g x ∂μ",
"tactic": "simp_rw [Pi.add_apply]"
},
{
"state_after": "case h.hf\nα : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\ni : Set α\nhi : MeasurableSet i\n⊢ IntegrableOn (fun x => f x) Set.univ\n\ncase h.hg\nα : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\ni : Set α\nhi : MeasurableSet i\n⊢ IntegrableOn (fun x => g x) Set.univ",
"state_before": "case h\nα : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\ni : Set α\nhi : MeasurableSet i\n⊢ (∫ (x : α) in i, f x + g x ∂μ) = (∫ (x : α) in i, f x ∂μ) + ∫ (x : α) in i, g x ∂μ",
"tactic": "rw [integral_add] <;> rw [← integrableOn_univ]"
},
{
"state_after": "no goals",
"state_before": "case h.hf\nα : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\ni : Set α\nhi : MeasurableSet i\n⊢ IntegrableOn (fun x => f x) Set.univ",
"tactic": "exact hf.integrableOn.restrict MeasurableSet.univ"
},
{
"state_after": "no goals",
"state_before": "case h.hg\nα : Type u_2\nβ : Type ?u.30911\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf g : α → E\nhf : Integrable f\nhg : Integrable g\ni : Set α\nhi : MeasurableSet i\n⊢ IntegrableOn (fun x => g x) Set.univ",
"tactic": "exact hg.integrableOn.restrict MeasurableSet.univ"
}
] |
[
96,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Logic/Unique.lean
|
unique_iff_exists_unique
|
[] |
[
70,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
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