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/CategoryTheory/Endomorphism.lean
|
CategoryTheory.End.smul_right
|
[] |
[
99,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
Submodule.coe_isComplEquivProj_symm_apply
|
[] |
[
364,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.MemRightTransversals.toEquiv_apply
|
[
{
"state_after": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) → G\nhf : ∀ (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq : Quotient (QuotientGroup.rightRel H)\n⊢ ↑(toEquiv (_ : (Set.range fun q => f q) ∈ rightTransversals ↑H)) q =\n { val := f q, property := (_ : ∃ y, (fun q => f q) y = f q) }",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) → G\nhf : ∀ (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq : Quotient (QuotientGroup.rightRel H)\n⊢ ↑(↑(toEquiv (_ : (Set.range fun q => f q) ∈ rightTransversals ↑H)) q) = f q",
"tactic": "refine' (Subtype.ext_iff.mp _).trans (Subtype.coe_mk (f q) ⟨q, rfl⟩)"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) → G\nhf : ∀ (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq : Quotient (QuotientGroup.rightRel H)\n⊢ ↑(toEquiv (_ : (Set.range fun q => f q) ∈ rightTransversals ↑H)) q =\n { val := f q, property := (_ : ∃ y, (fun q => f q) y = f q) }",
"tactic": "exact (toEquiv (range_mem_rightTransversals hf)).apply_eq_iff_eq_symm_apply.mpr (hf q).symm"
}
] |
[
410,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.single_pow
|
[
{
"state_after": "k : Type u₁\nG : Type u₂\nR : Type ?u.366813\ninst✝¹ : Semiring k\ninst✝ : Monoid G\na : G\nb : k\n⊢ 1 = single 1 1",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.366813\ninst✝¹ : Semiring k\ninst✝ : Monoid G\na : G\nb : k\n⊢ single a b ^ 0 = single (a ^ 0) (b ^ 0)",
"tactic": "simp only [pow_zero]"
},
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.366813\ninst✝¹ : Semiring k\ninst✝ : Monoid G\na : G\nb : k\n⊢ 1 = single 1 1",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "k : Type u₁\nG : Type u₂\nR : Type ?u.366813\ninst✝¹ : Semiring k\ninst✝ : Monoid G\na : G\nb : k\nn : ℕ\n⊢ single a b ^ (n + 1) = single (a ^ (n + 1)) (b ^ (n + 1))",
"tactic": "simp only [pow_succ, single_pow n, single_mul_single]"
}
] |
[
469,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
1
] |
Mathlib/CategoryTheory/Endofunctor/Algebra.lean
|
CategoryTheory.Endofunctor.Adjunction.Algebra.homEquiv_naturality_str
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nF G : C ⥤ C\nadj : F ⊣ G\nA₁ A₂ : Algebra F\nf : A₁ ⟶ A₂\n⊢ ↑(Adjunction.homEquiv adj A₁.a A₁.a) A₁.str ≫ G.map f.f = f.f ≫ ↑(Adjunction.homEquiv adj A₂.a A₂.a) A₂.str",
"tactic": "rw [← Adjunction.homEquiv_naturality_right, ← Adjunction.homEquiv_naturality_left, f.h]"
}
] |
[
459,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
1
] |
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
Complex.exp_eq_exp_ℂ
|
[
{
"state_after": "x : ℂ\n⊢ exp x = _root_.exp ℂ x",
"state_before": "⊢ exp = _root_.exp ℂ",
"tactic": "refine' funext fun x => _"
},
{
"state_after": "x : ℂ\n⊢ CauSeq.lim (exp' x) = (fun x => ∑' (n : ℕ), x ^ n / ↑n !) x",
"state_before": "x : ℂ\n⊢ exp x = _root_.exp ℂ x",
"tactic": "rw [Complex.exp, exp_eq_tsum_div]"
},
{
"state_after": "x : ℂ\nthis : CauSeq.IsComplete ℂ norm\n⊢ CauSeq.lim (exp' x) = (fun x => ∑' (n : ℕ), x ^ n / ↑n !) x",
"state_before": "x : ℂ\n⊢ CauSeq.lim (exp' x) = (fun x => ∑' (n : ℕ), x ^ n / ↑n !) x",
"tactic": "have : CauSeq.IsComplete ℂ norm := Complex.instIsComplete"
},
{
"state_after": "no goals",
"state_before": "x : ℂ\nthis : CauSeq.IsComplete ℂ norm\n⊢ CauSeq.lim (exp' x) = (fun x => ∑' (n : ℕ), x ^ n / ↑n !) x",
"tactic": "exact tendsto_nhds_unique x.exp'.tendsto_limit (expSeries_div_summable ℝ x).hasSum.tendsto_sum_nat"
}
] |
[
226,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Fintype.card_eq
|
[] |
[
212,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/RingTheory/IsTensorProduct.lean
|
IsBaseChange.ofEquiv
|
[
{
"state_after": "case h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q\ninst✝ : Module S Q\ne : M ≃ₗ[R] N\n⊢ ∀ (Q : Type (max v₁ v₂ u_1)) [inst : AddCommMonoid Q] [inst_1 : Module R Q] [inst_2 : Module R Q]\n [inst_3 : IsScalarTower R R Q] (g : M →ₗ[R] Q), ∃! g', LinearMap.comp (↑R g') ↑e = g",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q\ninst✝ : Module S Q\ne : M ≃ₗ[R] N\n⊢ IsBaseChange R ↑e",
"tactic": "apply IsBaseChange.of_lift_unique"
},
{
"state_after": "case h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\n⊢ ∃! g', LinearMap.comp (↑R g') ↑e = g",
"state_before": "case h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q\ninst✝ : Module S Q\ne : M ≃ₗ[R] N\n⊢ ∀ (Q : Type (max v₁ v₂ u_1)) [inst : AddCommMonoid Q] [inst_1 : Module R Q] [inst_2 : Module R Q]\n [inst_3 : IsScalarTower R R Q] (g : M →ₗ[R] Q), ∃! g', LinearMap.comp (↑R g') ↑e = g",
"tactic": "intro Q I₁ I₂ I₃ I₄ g"
},
{
"state_after": "case h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nthis : I₂ = I₃\n⊢ ∃! g', LinearMap.comp (↑R g') ↑e = g",
"state_before": "case h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\n⊢ ∃! g', LinearMap.comp (↑R g') ↑e = g",
"tactic": "have : I₂ = I₃ := by\n ext r q\n show (by let _ := I₂; exact r • q) = (by let _ := I₃; exact r • q)\n dsimp\n rw [← one_smul R q, smul_smul, ← @smul_assoc _ _ _ (id _) (id _) (id _) I₄, smul_eq_mul,\n mul_one]"
},
{
"state_after": "case h.refl\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ : Module R Q\ng : M →ₗ[R] Q\nI₄ : IsScalarTower R R Q\n⊢ ∃! g', LinearMap.comp (↑R g') ↑e = g",
"state_before": "case h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nthis : I₂ = I₃\n⊢ ∃! g', LinearMap.comp (↑R g') ↑e = g",
"tactic": "cases this"
},
{
"state_after": "case h.refl\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ : Module R Q\ng : M →ₗ[R] Q\nI₄ : IsScalarTower R R Q\n⊢ ∀ (y : N →ₗ[R] Q), (fun g' => LinearMap.comp (↑R g') ↑e = g) y → y = LinearMap.comp g ↑(LinearEquiv.symm e)",
"state_before": "case h.refl\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ : Module R Q\ng : M →ₗ[R] Q\nI₄ : IsScalarTower R R Q\n⊢ ∃! g', LinearMap.comp (↑R g') ↑e = g",
"tactic": "refine'\n ⟨g.comp e.symm.toLinearMap, by\n ext\n simp, _⟩"
},
{
"state_after": "case h.refl\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ : Module R Q\nI₄ : IsScalarTower R R Q\ny : N →ₗ[R] Q\n⊢ y = LinearMap.comp (LinearMap.comp (↑R y) ↑e) ↑(LinearEquiv.symm e)",
"state_before": "case h.refl\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ : Module R Q\ng : M →ₗ[R] Q\nI₄ : IsScalarTower R R Q\n⊢ ∀ (y : N →ₗ[R] Q), (fun g' => LinearMap.comp (↑R g') ↑e = g) y → y = LinearMap.comp g ↑(LinearEquiv.symm e)",
"tactic": "rintro y (rfl : _ = _)"
},
{
"state_after": "case h.refl.h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ : Module R Q\nI₄ : IsScalarTower R R Q\ny : N →ₗ[R] Q\nx✝ : N\n⊢ ↑y x✝ = ↑(LinearMap.comp (LinearMap.comp (↑R y) ↑e) ↑(LinearEquiv.symm e)) x✝",
"state_before": "case h.refl\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ : Module R Q\nI₄ : IsScalarTower R R Q\ny : N →ₗ[R] Q\n⊢ y = LinearMap.comp (LinearMap.comp (↑R y) ↑e) ↑(LinearEquiv.symm e)",
"tactic": "ext"
},
{
"state_after": "no goals",
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"tactic": "simp"
},
{
"state_after": "case smul.h.h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\n⊢ SMul.smul r q = SMul.smul r q",
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"tactic": "ext r q"
},
{
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"state_before": "case smul.h.h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\n⊢ SMul.smul r q = SMul.smul r q",
"tactic": "show (by let _ := I₂; exact r • q) = (by let _ := I₃; exact r • q)"
},
{
"state_after": "case smul.h.h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\n⊢ r • q = r • q",
"state_before": "case smul.h.h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\n⊢ (let x := I₂;\n r • q) =\n let x := I₃;\n r • q",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case smul.h.h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\n⊢ r • q = r • q",
"tactic": "rw [← one_smul R q, smul_smul, ← @smul_assoc _ _ _ (id _) (id _) (id _) I₄, smul_eq_mul,\n mul_one]"
},
{
"state_after": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\nx✝ : Module R Q := I₂\n⊢ ?m.500189",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\n⊢ ?m.500189",
"tactic": "let _ := I₂"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\nx✝ : Module R Q := I₂\n⊢ ?m.500189",
"tactic": "exact r • q"
},
{
"state_after": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\nx✝ : Module R Q := I₃\n⊢ Q",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\n⊢ Q",
"tactic": "let _ := I₃"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ I₃ : Module R Q\nI₄ : IsScalarTower R R Q\ng : M →ₗ[R] Q\nr : R\nq : Q\nx✝ : Module R Q := I₃\n⊢ Q",
"tactic": "exact r • q"
},
{
"state_after": "case h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ : Module R Q\ng : M →ₗ[R] Q\nI₄ : IsScalarTower R R Q\nx✝ : M\n⊢ ↑(LinearMap.comp (↑R (LinearMap.comp g ↑(LinearEquiv.symm e))) ↑e) x✝ = ↑g x✝",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ : Module R Q\ng : M →ₗ[R] Q\nI₄ : IsScalarTower R R Q\n⊢ (fun g' => LinearMap.comp (↑R g') ↑e = g) (LinearMap.comp g ↑(LinearEquiv.symm e))",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module S N\ninst✝⁴ : IsScalarTower R S N\nf : M →ₗ[R] N\nh : IsBaseChange S f\nP : Type ?u.497103\nQ✝ : Type ?u.497106\ninst✝³ : AddCommMonoid P\ninst✝² : Module R P\ninst✝¹ : AddCommMonoid Q✝\ninst✝ : Module S Q✝\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\nI₁ : AddCommMonoid Q\nI₂ : Module R Q\ng : M →ₗ[R] Q\nI₄ : IsScalarTower R R Q\nx✝ : M\n⊢ ↑(LinearMap.comp (↑R (LinearMap.comp g ↑(LinearEquiv.symm e))) ↑e) x✝ = ↑g x✝",
"tactic": "simp"
}
] |
[
329,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Topology/Sequences.lean
|
IsSeqCompact.subseq_of_frequently_in
|
[] |
[
273,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
|
MeasureTheory.Measure.tendsto_add_haar_inter_smul_one_of_density_one_aux
|
[
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)",
"tactic": "have L' : Tendsto (fun r : ℝ => μ (sᶜ ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) :=\n tendsto_add_haar_inter_smul_zero_of_density_zero μ (sᶜ) x L t ht h''t"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)",
"tactic": "have L'' : Tendsto (fun r : ℝ => μ ({x} + r • t) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1) := by\n apply tendsto_const_nhds.congr' _\n filter_upwards [self_mem_nhdsWithin]\n rintro r (rpos : 0 < r)\n rw [add_haar_singleton_add_smul_div_singleton_add_smul μ rpos.ne', ENNReal.div_self h't h''t]"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 (1 - 0))\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)",
"tactic": "have := ENNReal.Tendsto.sub L'' L' (Or.inl ENNReal.one_ne_top)"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 (1 - 0))\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)",
"tactic": "simp only [tsub_zero] at this"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\n⊢ (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) =ᶠ[𝓝[Ioi 0] 0]\n fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)",
"tactic": "apply this.congr' _"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\n⊢ ∀ (a : ℝ),\n a ∈ Ioi 0 →\n ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t) =\n ↑↑μ (s ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\n⊢ (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) =ᶠ[𝓝[Ioi 0] 0]\n fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t) - ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t) =\n ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\n⊢ ∀ (a : ℝ),\n a ∈ Ioi 0 →\n ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t) =\n ↑↑μ (s ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)",
"tactic": "rintro r (rpos : 0 < r)"
},
{
"state_after": "case h.refine'_1\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ ({x} + r • t) ≠ 0\n\ncase h.refine'_2\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ ({x} + r • t) ≠ ⊤",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t) - ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t) =\n ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)",
"tactic": "refine' I ({x} + r • t) s _ _ hs"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\n⊢ ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u",
"tactic": "intro u v uzero utop vmeas"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ ↑↑μ u * (↑↑μ u)⁻¹ - ↑↑μ (vᶜ ∩ u) * (↑↑μ u)⁻¹ = ↑↑μ (v ∩ u) * (↑↑μ u)⁻¹",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u",
"tactic": "simp_rw [div_eq_mul_inv]"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ (↑↑μ u - ↑↑μ (vᶜ ∩ u)) * (↑↑μ u)⁻¹ = ↑↑μ (v ∩ u) * (↑↑μ u)⁻¹\n\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ 0 < ↑↑μ (vᶜ ∩ u) → ↑↑μ (vᶜ ∩ u) < ↑↑μ u → (↑↑μ u)⁻¹ ≠ ⊤",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ ↑↑μ u * (↑↑μ u)⁻¹ - ↑↑μ (vᶜ ∩ u) * (↑↑μ u)⁻¹ = ↑↑μ (v ∩ u) * (↑↑μ u)⁻¹",
"tactic": "rw [← ENNReal.sub_mul]"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ 0 < ↑↑μ (vᶜ ∩ u) → ↑↑μ (vᶜ ∩ u) < ↑↑μ u → (↑↑μ u)⁻¹ ≠ ⊤\n\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ (↑↑μ u - ↑↑μ (vᶜ ∩ u)) * (↑↑μ u)⁻¹ = ↑↑μ (v ∩ u) * (↑↑μ u)⁻¹",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ (↑↑μ u - ↑↑μ (vᶜ ∩ u)) * (↑↑μ u)⁻¹ = ↑↑μ (v ∩ u) * (↑↑μ u)⁻¹\n\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ 0 < ↑↑μ (vᶜ ∩ u) → ↑↑μ (vᶜ ∩ u) < ↑↑μ u → (↑↑μ u)⁻¹ ≠ ⊤",
"tactic": "swap"
},
{
"state_after": "case e_a\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ ↑↑μ u - ↑↑μ (vᶜ ∩ u) = ↑↑μ (v ∩ u)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ (↑↑μ u - ↑↑μ (vᶜ ∩ u)) * (↑↑μ u)⁻¹ = ↑↑μ (v ∩ u) * (↑↑μ u)⁻¹",
"tactic": "congr 1"
},
{
"state_after": "case e_a\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ ↑↑μ (v ∩ u) + ↑↑μ (vᶜ ∩ u) = ↑↑μ u",
"state_before": "case e_a\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ ↑↑μ u - ↑↑μ (vᶜ ∩ u) = ↑↑μ (v ∩ u)",
"tactic": "apply\n ENNReal.sub_eq_of_add_eq (ne_top_of_le_ne_top utop (measure_mono (inter_subset_right _ _)))"
},
{
"state_after": "case e_a\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ ↑↑μ (u ∩ v) + ↑↑μ (u ∩ vᶜ) = ↑↑μ u",
"state_before": "case e_a\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ ↑↑μ (v ∩ u) + ↑↑μ (vᶜ ∩ u) = ↑↑μ u",
"tactic": "rw [inter_comm _ u, inter_comm _ u]"
},
{
"state_after": "no goals",
"state_before": "case e_a\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ ↑↑μ (u ∩ v) + ↑↑μ (u ∩ vᶜ) = ↑↑μ u",
"tactic": "exact measure_inter_add_diff u vmeas"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nu v : Set E\nuzero : ↑↑μ u ≠ 0\nutop : ↑↑μ u ≠ ⊤\nvmeas : MeasurableSet v\n⊢ 0 < ↑↑μ (vᶜ ∩ u) → ↑↑μ (vᶜ ∩ u) < ↑↑μ u → (↑↑μ u)⁻¹ ≠ ⊤",
"tactic": "simp only [uzero, ENNReal.inv_eq_top, imp_true_iff, Ne.def, not_false_iff]"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nA : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nB :\n Tendsto (fun a => ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a) - ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a))\n (𝓝[Ioi 0] 0) (𝓝 (1 - 1))\n⊢ Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nA : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\n⊢ Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)",
"tactic": "have B := ENNReal.Tendsto.sub A h (Or.inl ENNReal.one_ne_top)"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nA : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nB :\n Tendsto (fun a => ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a) - ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nA : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nB :\n Tendsto (fun a => ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a) - ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a))\n (𝓝[Ioi 0] 0) (𝓝 (1 - 1))\n⊢ Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)",
"tactic": "simp only [tsub_self] at B"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nA : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nB :\n Tendsto (fun a => ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a) - ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ (fun a =>\n ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a) - ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a)) =ᶠ[𝓝[Ioi 0] 0]\n fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nA : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nB :\n Tendsto (fun a => ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a) - ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)",
"tactic": "apply B.congr' _"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nA : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nB :\n Tendsto (fun a => ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a) - ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ ∀ (a : ℝ),\n a ∈ Ioi 0 →\n ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a) - ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a) =\n ↑↑μ (sᶜ ∩ closedBall x a) / ↑↑μ (closedBall x a)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nA : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nB :\n Tendsto (fun a => ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a) - ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ (fun a =>\n ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a) - ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a)) =ᶠ[𝓝[Ioi 0] 0]\n fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
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"tactic": "rintro r (rpos : 0 < r)"
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"tactic": "convert I (closedBall x r) (sᶜ) (measure_closedBall_pos μ _ rpos).ne'\n measure_closedBall_lt_top.ne hs.compl"
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"tactic": "rw [compl_compl]"
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"tactic": "apply tendsto_const_nhds.congr' _"
},
{
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"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
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"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\n⊢ ∀ (a : ℝ), a ∈ Ioi 0 → 1 = ↑↑μ (closedBall x a) / ↑↑μ (closedBall x a)",
"tactic": "intro r hr"
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"tactic": "rw [div_eq_mul_inv, ENNReal.mul_inv_cancel]"
},
{
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"tactic": "exact (measure_closedBall_pos μ _ hr).ne'"
},
{
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"state_before": "case h.ht\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nr : ℝ\nhr : r ∈ Ioi 0\n⊢ ↑↑μ (closedBall x r) ≠ ⊤",
"tactic": "exact measure_closedBall_lt_top.ne"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ (fun x => 1) =ᶠ[𝓝[Ioi 0] 0] fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)",
"tactic": "apply tendsto_const_nhds.congr' _"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ ∀ (a : ℝ), a ∈ Ioi 0 → 1 = ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ (fun x => 1) =ᶠ[𝓝[Ioi 0] 0] fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ 1 = ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ ∀ (a : ℝ), a ∈ Ioi 0 → 1 = ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t)",
"tactic": "rintro r (rpos : 0 < r)"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ 1 = ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)",
"tactic": "rw [add_haar_singleton_add_smul_div_singleton_add_smul μ rpos.ne', ENNReal.div_self h't h''t]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_1\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ ({x} + r • t) ≠ 0",
"tactic": "simp only [h't, abs_of_nonneg rpos.le, pow_pos rpos, add_haar_smul, image_add_left,\n ENNReal.ofReal_eq_zero, not_le, or_false_iff, Ne.def, measure_preimage_add, abs_pow,\n singleton_add, mul_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2255315\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)\nt : Set E\nht : MeasurableSet t\nh't : ↑↑μ t ≠ 0\nh''t : ↑↑μ t ≠ ⊤\nI :\n ∀ (u v : Set E), ↑↑μ u ≠ 0 → ↑↑μ u ≠ ⊤ → MeasurableSet v → ↑↑μ u / ↑↑μ u - ↑↑μ (vᶜ ∩ u) / ↑↑μ u = ↑↑μ (v ∩ u) / ↑↑μ u\nL : Tendsto (fun r => ↑↑μ (sᶜ ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nL' : Tendsto (fun r => ↑↑μ (sᶜ ∩ ({x} + r • t)) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 0)\nL'' : Tendsto (fun r => ↑↑μ ({x} + r • t) / ↑↑μ ({x} + r • t)) (𝓝[Ioi 0] 0) (𝓝 1)\nthis :\n Tendsto (fun a => ↑↑μ ({x} + a • t) / ↑↑μ ({x} + a • t) - ↑↑μ (sᶜ ∩ ({x} + a • t)) / ↑↑μ ({x} + a • t)) (𝓝[Ioi 0] 0)\n (𝓝 1)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ ({x} + r • t) ≠ ⊤",
"tactic": "simp [h''t, ENNReal.ofReal_ne_top, add_haar_smul, image_add_left, ENNReal.mul_eq_top,\n Ne.def, not_false_iff, measure_preimage_add, singleton_add, and_false_iff, false_and_iff,\n or_self_iff]"
}
] |
[
804,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
755,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.zero_mod
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.258355\nβ : Type ?u.258358\nγ : Type ?u.258361\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nb : Ordinal\n⊢ 0 % b = 0",
"tactic": "simp only [mod_def, zero_div, mul_zero, sub_self]"
}
] |
[
1044,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1044,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.iSup_union
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.436212\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.436221\nι : Type ?u.436224\nκ : Type ?u.436227\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nf : α → β\ns t : Finset α\n⊢ (⨆ (x : α) (_ : x ∈ s ∪ t), f x) = (⨆ (x : α) (_ : x ∈ s), f x) ⊔ ⨆ (x : α) (_ : x ∈ t), f x",
"tactic": "simp [iSup_or, iSup_sup_eq]"
}
] |
[
1949,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1948,
1
] |
Mathlib/Analysis/Convex/Slope.lean
|
ConvexOn.strict_mono_of_lt
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\n⊢ f u < f v",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\n⊢ StrictMonoOn f (s ∩ Set.Ici y)",
"tactic": "intro u hu v hv huv"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\n⊢ f u < f v",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\n⊢ f u < f v",
"tactic": "have step1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z := by\n intros z hz\n refine hf.lt_right_of_left_lt hx hz.1 ?_ hxy'\n rw [openSegment_eq_Ioo (hxy.trans hz.2)]\n exact ⟨hxy, hz.2⟩"
},
{
"state_after": "case inl\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu : y ∈ s ∩ Set.Ici y\nhuv : y < v\n⊢ f y < f v\n\ncase inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ f u < f v",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\n⊢ f u < f v",
"tactic": "rcases eq_or_lt_of_le hu.2 with (rfl | hu2)"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nz : 𝕜\nhz : z ∈ s ∩ Set.Ioi y\n⊢ f y < f z",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\n⊢ ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z",
"tactic": "intros z hz"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nz : 𝕜\nhz : z ∈ s ∩ Set.Ioi y\n⊢ y ∈ openSegment 𝕜 x z",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nz : 𝕜\nhz : z ∈ s ∩ Set.Ioi y\n⊢ f y < f z",
"tactic": "refine hf.lt_right_of_left_lt hx hz.1 ?_ hxy'"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nz : 𝕜\nhz : z ∈ s ∩ Set.Ioi y\n⊢ y ∈ Set.Ioo x z",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nz : 𝕜\nhz : z ∈ s ∩ Set.Ioi y\n⊢ y ∈ openSegment 𝕜 x z",
"tactic": "rw [openSegment_eq_Ioo (hxy.trans hz.2)]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nz : 𝕜\nhz : z ∈ s ∩ Set.Ioi y\n⊢ y ∈ Set.Ioo x z",
"tactic": "exact ⟨hxy, hz.2⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu : y ∈ s ∩ Set.Ici y\nhuv : y < v\n⊢ f y < f v",
"tactic": "exact step1 ⟨hv.1, huv⟩"
},
{
"state_after": "case inr.refine'_1\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ y ∈ s\n\ncase inr.refine'_2\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ u ∈ openSegment 𝕜 y v",
"state_before": "case inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ f u < f v",
"tactic": "refine' hf.lt_right_of_left_lt _ hv.1 _ (step1 ⟨hu.1, hu2⟩)"
},
{
"state_after": "case inr.refine'_1.a\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ y ∈ segment 𝕜 x u",
"state_before": "case inr.refine'_1\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ y ∈ s",
"tactic": "apply hf.1.segment_subset hx hu.1"
},
{
"state_after": "case inr.refine'_1.a\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ y ∈ Set.Icc x u",
"state_before": "case inr.refine'_1.a\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ y ∈ segment 𝕜 x u",
"tactic": "rw [segment_eq_Icc (hxy.le.trans hu.2)]"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_1.a\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ y ∈ Set.Icc x u",
"tactic": "exact ⟨hxy.le, hu.2⟩"
},
{
"state_after": "case inr.refine'_2\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ u ∈ Set.Ioo y v",
"state_before": "case inr.refine'_2\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ u ∈ openSegment 𝕜 y v",
"tactic": "rw [openSegment_eq_Ioo (hu2.trans huv)]"
},
{
"state_after": "no goals",
"state_before": "case inr.refine'_2\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z\nhu2 : y < u\n⊢ u ∈ Set.Ioo y v",
"tactic": "exact ⟨hu2, huv⟩"
}
] |
[
365,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.centralizer_eq_top_iff_subset
|
[] |
[
1416,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1415,
1
] |
Mathlib/Data/MvPolynomial/Expand.lean
|
MvPolynomial.expand_monomial
|
[] |
[
55,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.NullMeasurableSet.biUnion_decode₂
|
[] |
[
157,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
11
] |
Mathlib/Data/Finset/PImage.lean
|
Part.toFinset_some
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.528\na : α\ninst✝ : Decidable (some a).Dom\n⊢ toFinset (some a) = {a}",
"tactic": "simp [toFinset]"
}
] |
[
48,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.logb_div
|
[
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ logb b (x / y) = logb b x - logb b y",
"tactic": "simp_rw [logb, log_div hx hy, sub_div]"
}
] |
[
73,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/GradedMonoid.lean
|
SetLike.homogeneous_coe
|
[] |
[
686,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
685,
1
] |
Mathlib/Data/PNat/Prime.lean
|
PNat.Prime.one_lt
|
[] |
[
122,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Topology/MetricSpace/CantorScheme.lean
|
CantorScheme.Disjoint.map_injective
|
[
{
"state_after": "case mk.mk\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\n⊢ { val := x, property := hx } = { val := y, property := hy }",
"state_before": "β : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\n⊢ Injective (inducedMap A).snd",
"tactic": "rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy"
},
{
"state_after": "case mk.mk\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\n⊢ res ((fun a => ↑a) { val := x, property := hx }) = res ((fun a => ↑a) { val := y, property := hy })",
"state_before": "case mk.mk\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\n⊢ { val := x, property := hx } = { val := y, property := hy }",
"tactic": "refine' Subtype.coe_injective (res_injective _)"
},
{
"state_after": "case mk.mk\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\n⊢ res x = res y",
"state_before": "case mk.mk\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\n⊢ res ((fun a => ↑a) { val := x, property := hx }) = res ((fun a => ↑a) { val := y, property := hy })",
"tactic": "dsimp"
},
{
"state_after": "case mk.mk.h\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\n⊢ res x n = res y n",
"state_before": "case mk.mk\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\n⊢ res x = res y",
"tactic": "ext n : 1"
},
{
"state_after": "case mk.mk.h.zero\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\n⊢ res x Nat.zero = res y Nat.zero\n\ncase mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\n⊢ res x (Nat.succ n) = res y (Nat.succ n)",
"state_before": "case mk.mk.h\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\n⊢ res x n = res y n",
"tactic": "induction' n with n ih"
},
{
"state_after": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\n⊢ x n = y n ∧ res x n = res y n",
"state_before": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\n⊢ res x (Nat.succ n) = res y (Nat.succ n)",
"tactic": "simp only [res_succ, cons.injEq]"
},
{
"state_after": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\n⊢ x n = y n",
"state_before": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\n⊢ x n = y n ∧ res x n = res y n",
"tactic": "refine' ⟨_, ih⟩"
},
{
"state_after": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ ¬CantorScheme.Disjoint A",
"state_before": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\n⊢ x n = y n",
"tactic": "contrapose hA"
},
{
"state_after": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ ∃ x x_1 x_2, ¬x_1 = x_2 ∧ ¬_root_.Disjoint (A (x_1 :: x)) (A (x_2 :: x))",
"state_before": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ ¬CantorScheme.Disjoint A",
"tactic": "simp only [CantorScheme.Disjoint, _root_.Pairwise, Ne.def, not_forall, exists_prop]"
},
{
"state_after": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ ¬_root_.Disjoint (A (x n :: res x n)) (A (y n :: res x n))",
"state_before": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ ∃ x x_1 x_2, ¬x_1 = x_2 ∧ ¬_root_.Disjoint (A (x_1 :: x)) (A (x_2 :: x))",
"tactic": "refine' ⟨res x n, _, _, hA, _⟩"
},
{
"state_after": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ ∃ x_1, x_1 ∈ A (x n :: res x n) ∧ x_1 ∈ A (y n :: res x n)",
"state_before": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ ¬_root_.Disjoint (A (x n :: res x n)) (A (y n :: res x n))",
"tactic": "rw [not_disjoint_iff]"
},
{
"state_after": "case mk.mk.h.succ.refine'_1\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ Sigma.snd (inducedMap A) { val := x, property := hx } ∈ A (x n :: res x n)\n\ncase mk.mk.h.succ.refine'_2\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ Sigma.snd (inducedMap A) { val := x, property := hx } ∈ A (y n :: res x n)",
"state_before": "case mk.mk.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ ∃ x_1, x_1 ∈ A (x n :: res x n) ∧ x_1 ∈ A (y n :: res x n)",
"tactic": "refine' ⟨(inducedMap A).2 ⟨x, hx⟩, _, _⟩"
},
{
"state_after": "case mk.mk.h.succ.refine'_2\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ Sigma.snd (inducedMap A) { val := y, property := hy } ∈ A (res y (Nat.succ n))",
"state_before": "case mk.mk.h.succ.refine'_2\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ Sigma.snd (inducedMap A) { val := x, property := hx } ∈ A (y n :: res x n)",
"tactic": "rw [hxy, ih, ← res_succ]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.h.succ.refine'_2\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ Sigma.snd (inducedMap A) { val := y, property := hy } ∈ A (res y (Nat.succ n))",
"tactic": "apply map_mem"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.h.zero\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nhA : CantorScheme.Disjoint A\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\n⊢ res x Nat.zero = res y Nat.zero",
"tactic": "simp"
},
{
"state_after": "case mk.mk.h.succ.refine'_1\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ Sigma.snd (inducedMap A) { val := x, property := hx } ∈ A (res x (Nat.succ n))",
"state_before": "case mk.mk.h.succ.refine'_1\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ Sigma.snd (inducedMap A) { val := x, property := hx } ∈ A (x n :: res x n)",
"tactic": "rw [← res_succ]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.h.succ.refine'_1\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx : ℕ → β\nhx : x ∈ (inducedMap A).fst\ny : ℕ → β\nhy : y ∈ (inducedMap A).fst\nhxy : Sigma.snd (inducedMap A) { val := x, property := hx } = Sigma.snd (inducedMap A) { val := y, property := hy }\nn : ℕ\nih : res x n = res y n\nhA : ¬x n = y n\n⊢ Sigma.snd (inducedMap A) { val := x, property := hx } ∈ A (res x (Nat.succ n))",
"tactic": "apply map_mem"
}
] |
[
117,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Ico.coe_eq_zero
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : OrderedSemiring α\ninst✝ : Nontrivial α\nx : ↑(Ico 0 1)\n⊢ x = 0 ↔ ↑x = 0",
"state_before": "α : Type u_1\ninst✝¹ : OrderedSemiring α\ninst✝ : Nontrivial α\nx : ↑(Ico 0 1)\n⊢ ↑x = 0 ↔ x = 0",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : OrderedSemiring α\ninst✝ : Nontrivial α\nx : ↑(Ico 0 1)\n⊢ x = 0 ↔ ↑x = 0",
"tactic": "exact Subtype.ext_iff"
}
] |
[
204,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Localization.r_eq_r'
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type ?u.37854\ninst✝¹ : CommMonoid N\nP : Type ?u.37860\ninst✝ : CommMonoid P\nx✝ : { x // x ∈ S }\n⊢ ↑1 * (↑(↑x✝, x✝).snd * 1.fst) = ↑1 * (↑1.snd * (↑x✝, x✝).fst)",
"tactic": "simp"
},
{
"state_after": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type ?u.37854\ninst✝¹ : CommMonoid N\nP : Type ?u.37860\ninst✝ : CommMonoid P\nb : Con (M × { x // x ∈ S })\nH : b ∈ {c | ∀ (y : { x // x ∈ S }), ↑c 1 (↑y, y)}\nx✝² x✝¹ : M × { x // x ∈ S }\np : M\nq : { x // x ∈ S }\nx : M\ny : { x // x ∈ S }\nx✝ : ↑(r' S) (p, q) (x, y)\nt : { x // x ∈ S }\nht : ↑t * (↑(x, y).snd * (p, q).fst) = ↑t * (↑(p, q).snd * (x, y).fst)\n⊢ ↑b (1 * (p, q)) (1 * (x, y))",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type ?u.37854\ninst✝¹ : CommMonoid N\nP : Type ?u.37860\ninst✝ : CommMonoid P\nb : Con (M × { x // x ∈ S })\nH : b ∈ {c | ∀ (y : { x // x ∈ S }), ↑c 1 (↑y, y)}\nx✝² x✝¹ : M × { x // x ∈ S }\np : M\nq : { x // x ∈ S }\nx : M\ny : { x // x ∈ S }\nx✝ : ↑(r' S) (p, q) (x, y)\nt : { x // x ∈ S }\nht : ↑t * (↑(x, y).snd * (p, q).fst) = ↑t * (↑(p, q).snd * (x, y).fst)\n⊢ ↑b (p, q) (x, y)",
"tactic": "rw [← one_mul (p, q), ← one_mul (x, y)]"
},
{
"state_after": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type ?u.37854\ninst✝¹ : CommMonoid N\nP : Type ?u.37860\ninst✝ : CommMonoid P\nb : Con (M × { x // x ∈ S })\nH : b ∈ {c | ∀ (y : { x // x ∈ S }), ↑c 1 (↑y, y)}\nx✝² x✝¹ : M × { x // x ∈ S }\np : M\nq : { x // x ∈ S }\nx : M\ny : { x // x ∈ S }\nx✝ : ↑(r' S) (p, q) (x, y)\nt : { x // x ∈ S }\nht : ↑t * (↑(x, y).snd * (p, q).fst) = ↑t * (↑(p, q).snd * (x, y).fst)\n⊢ ↑b ((↑(t * y), t * y) * (p, q)) (1 * (x, y))",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type ?u.37854\ninst✝¹ : CommMonoid N\nP : Type ?u.37860\ninst✝ : CommMonoid P\nb : Con (M × { x // x ∈ S })\nH : b ∈ {c | ∀ (y : { x // x ∈ S }), ↑c 1 (↑y, y)}\nx✝² x✝¹ : M × { x // x ∈ S }\np : M\nq : { x // x ∈ S }\nx : M\ny : { x // x ∈ S }\nx✝ : ↑(r' S) (p, q) (x, y)\nt : { x // x ∈ S }\nht : ↑t * (↑(x, y).snd * (p, q).fst) = ↑t * (↑(p, q).snd * (x, y).fst)\n⊢ ↑b (1 * (p, q)) (1 * (x, y))",
"tactic": "refine b.trans (b.mul (H (t * y)) (b.refl _)) ?_"
},
{
"state_after": "case h.e'_3\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type ?u.37854\ninst✝¹ : CommMonoid N\nP : Type ?u.37860\ninst✝ : CommMonoid P\nb : Con (M × { x // x ∈ S })\nH : b ∈ {c | ∀ (y : { x // x ∈ S }), ↑c 1 (↑y, y)}\nx✝² x✝¹ : M × { x // x ∈ S }\np : M\nq : { x // x ∈ S }\nx : M\ny : { x // x ∈ S }\nx✝ : ↑(r' S) (p, q) (x, y)\nt : { x // x ∈ S }\nht : ↑t * (↑(x, y).snd * (p, q).fst) = ↑t * (↑(p, q).snd * (x, y).fst)\n⊢ (↑(t * y), t * y) * (p, q) = (↑(t * q), t * q) * (x, y)",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type ?u.37854\ninst✝¹ : CommMonoid N\nP : Type ?u.37860\ninst✝ : CommMonoid P\nb : Con (M × { x // x ∈ S })\nH : b ∈ {c | ∀ (y : { x // x ∈ S }), ↑c 1 (↑y, y)}\nx✝² x✝¹ : M × { x // x ∈ S }\np : M\nq : { x // x ∈ S }\nx : M\ny : { x // x ∈ S }\nx✝ : ↑(r' S) (p, q) (x, y)\nt : { x // x ∈ S }\nht : ↑t * (↑(x, y).snd * (p, q).fst) = ↑t * (↑(p, q).snd * (x, y).fst)\n⊢ ↑b ((↑(t * y), t * y) * (p, q)) (1 * (x, y))",
"tactic": "convert b.symm (b.mul (H (t * q)) (b.refl (x, y))) using 1"
},
{
"state_after": "case h.e'_3\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type ?u.37854\ninst✝¹ : CommMonoid N\nP : Type ?u.37860\ninst✝ : CommMonoid P\nb : Con (M × { x // x ∈ S })\nH : b ∈ {c | ∀ (y : { x // x ∈ S }), ↑c 1 (↑y, y)}\nx✝² x✝¹ : M × { x // x ∈ S }\np : M\nq : { x // x ∈ S }\nx : M\ny : { x // x ∈ S }\nx✝ : ↑(r' S) (p, q) (x, y)\nt : { x // x ∈ S }\nht : ↑t * (↑y * p) = ↑t * (↑q * x)\n⊢ (↑t * ↑y * p, t * y * q) = (↑t * ↑q * x, t * q * y)",
"state_before": "case h.e'_3\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type ?u.37854\ninst✝¹ : CommMonoid N\nP : Type ?u.37860\ninst✝ : CommMonoid P\nb : Con (M × { x // x ∈ S })\nH : b ∈ {c | ∀ (y : { x // x ∈ S }), ↑c 1 (↑y, y)}\nx✝² x✝¹ : M × { x // x ∈ S }\np : M\nq : { x // x ∈ S }\nx : M\ny : { x // x ∈ S }\nx✝ : ↑(r' S) (p, q) (x, y)\nt : { x // x ∈ S }\nht : ↑t * (↑(x, y).snd * (p, q).fst) = ↑t * (↑(p, q).snd * (x, y).fst)\n⊢ (↑(t * y), t * y) * (p, q) = (↑(t * q), t * q) * (x, y)",
"tactic": "dsimp only [Prod.mk_mul_mk, Submonoid.coe_mul] at ht ⊢"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type ?u.37854\ninst✝¹ : CommMonoid N\nP : Type ?u.37860\ninst✝ : CommMonoid P\nb : Con (M × { x // x ∈ S })\nH : b ∈ {c | ∀ (y : { x // x ∈ S }), ↑c 1 (↑y, y)}\nx✝² x✝¹ : M × { x // x ∈ S }\np : M\nq : { x // x ∈ S }\nx : M\ny : { x // x ∈ S }\nx✝ : ↑(r' S) (p, q) (x, y)\nt : { x // x ∈ S }\nht : ↑t * (↑y * p) = ↑t * (↑q * x)\n⊢ (↑t * ↑y * p, t * y * q) = (↑t * ↑q * x, t * q * y)",
"tactic": "simp_rw [mul_assoc, ht, mul_comm y q]"
}
] |
[
198,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.sum_smul_vsub_const_eq_weightedVSubOfPoint_sub
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_3\nV : Type u_1\nP : Type u_4\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type ?u.101169\ns₂ : Finset ι₂\nw : ι → k\np₁ : ι → P\np₂ b : P\n⊢ ∑ i in s, w i • (p₁ i -ᵥ p₂) = ↑(weightedVSubOfPoint s p₁ b) w - (∑ i in s, w i) • (p₂ -ᵥ b)",
"tactic": "rw [sum_smul_vsub_eq_weightedVSubOfPoint_sub, weightedVSubOfPoint_apply_const]"
}
] |
[
198,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
MeasureTheory.Measure.dirac_prod_dirac
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.4493213\nβ : Type u_2\nβ' : Type ?u.4493219\nγ : Type ?u.4493222\nE : Type ?u.4493225\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\nx : α\ny : β\n⊢ Measure.prod (dirac x) (dirac y) = dirac (x, y)",
"tactic": "rw [prod_dirac, map_dirac measurable_prod_mk_right]"
}
] |
[
568,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.updateColumn_apply
|
[
{
"state_after": "case pos\nl : Type ?u.1220689\nm : Type u_2\nn : Type u_1\no : Type ?u.1220698\nm' : o → Type ?u.1220703\nn' : o → Type ?u.1220708\nR : Type ?u.1220711\nS : Type ?u.1220714\nα : Type v\nβ : Type w\nγ : Type ?u.1220721\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝ : DecidableEq n\nj' : n\nh : j' = j\n⊢ updateColumn M j c i j' = if j' = j then c i else M i j'\n\ncase neg\nl : Type ?u.1220689\nm : Type u_2\nn : Type u_1\no : Type ?u.1220698\nm' : o → Type ?u.1220703\nn' : o → Type ?u.1220708\nR : Type ?u.1220711\nS : Type ?u.1220714\nα : Type v\nβ : Type w\nγ : Type ?u.1220721\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝ : DecidableEq n\nj' : n\nh : ¬j' = j\n⊢ updateColumn M j c i j' = if j' = j then c i else M i j'",
"state_before": "l : Type ?u.1220689\nm : Type u_2\nn : Type u_1\no : Type ?u.1220698\nm' : o → Type ?u.1220703\nn' : o → Type ?u.1220708\nR : Type ?u.1220711\nS : Type ?u.1220714\nα : Type v\nβ : Type w\nγ : Type ?u.1220721\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝ : DecidableEq n\nj' : n\n⊢ updateColumn M j c i j' = if j' = j then c i else M i j'",
"tactic": "by_cases j' = j"
},
{
"state_after": "no goals",
"state_before": "case pos\nl : Type ?u.1220689\nm : Type u_2\nn : Type u_1\no : Type ?u.1220698\nm' : o → Type ?u.1220703\nn' : o → Type ?u.1220708\nR : Type ?u.1220711\nS : Type ?u.1220714\nα : Type v\nβ : Type w\nγ : Type ?u.1220721\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝ : DecidableEq n\nj' : n\nh : j' = j\n⊢ updateColumn M j c i j' = if j' = j then c i else M i j'",
"tactic": "rw [h, updateColumn_self, if_pos rfl]"
},
{
"state_after": "no goals",
"state_before": "case neg\nl : Type ?u.1220689\nm : Type u_2\nn : Type u_1\no : Type ?u.1220698\nm' : o → Type ?u.1220703\nn' : o → Type ?u.1220708\nR : Type ?u.1220711\nS : Type ?u.1220714\nα : Type v\nβ : Type w\nγ : Type ?u.1220721\nM : Matrix m n α\ni : m\nj : n\nb : n → α\nc : m → α\ninst✝ : DecidableEq n\nj' : n\nh : ¬j' = j\n⊢ updateColumn M j c i j' = if j' = j then c i else M i j'",
"tactic": "rw [updateColumn_ne h, if_neg h]"
}
] |
[
2784,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2780,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.get?_tail
|
[] |
[
279,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Topology/StoneCech.lean
|
stoneCechExtend_extends
|
[] |
[
280,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.eq_union_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.179008\nγ : Type ?u.179011\ninst✝ : DecidableEq α\ns t u : Multiset α\na b : α\nh : s ≤ t\n⊢ s ∪ t = t",
"tactic": "rw [union_comm, eq_union_left h]"
}
] |
[
1835,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1835,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.abs_re_le_abs
|
[
{
"state_after": "z : ℂ\n⊢ z.re ^ 2 ≤ z.re ^ 2 + z.im * z.im",
"state_before": "z : ℂ\n⊢ z.re ^ 2 ≤ ↑normSq z",
"tactic": "rw [normSq_apply, ← sq]"
},
{
"state_after": "no goals",
"state_before": "z : ℂ\n⊢ z.re ^ 2 ≤ z.re ^ 2 + z.im * z.im",
"tactic": "exact le_add_of_nonneg_right (mul_self_nonneg _)"
}
] |
[
1033,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1030,
1
] |
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_one
|
[] |
[
285,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
InfHom.id_apply
|
[] |
[
582,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
581,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.integral_diff
|
[
{
"state_after": "case hst\nα : Type u_1\nβ : Type ?u.7261\nE : Type u_2\nF : Type ?u.7267\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht : MeasurableSet t\nhfs : IntegrableOn f s\nhts : t ⊆ s\n⊢ Disjoint (s \\ t) t\n\ncase ht\nα : Type u_1\nβ : Type ?u.7261\nE : Type u_2\nF : Type ?u.7267\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht : MeasurableSet t\nhfs : IntegrableOn f s\nhts : t ⊆ s\n⊢ MeasurableSet t\n\ncase hfs\nα : Type u_1\nβ : Type ?u.7261\nE : Type u_2\nF : Type ?u.7267\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht : MeasurableSet t\nhfs : IntegrableOn f s\nhts : t ⊆ s\n⊢ IntegrableOn (fun x => f x) (s \\ t)\n\ncase hft\nα : Type u_1\nβ : Type ?u.7261\nE : Type u_2\nF : Type ?u.7267\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht : MeasurableSet t\nhfs : IntegrableOn f s\nhts : t ⊆ s\n⊢ IntegrableOn (fun x => f x) t",
"state_before": "α : Type u_1\nβ : Type ?u.7261\nE : Type u_2\nF : Type ?u.7267\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht : MeasurableSet t\nhfs : IntegrableOn f s\nhts : t ⊆ s\n⊢ (∫ (x : α) in s \\ t, f x ∂μ) = (∫ (x : α) in s, f x ∂μ) - ∫ (x : α) in t, f x ∂μ",
"tactic": "rw [eq_sub_iff_add_eq, ← integral_union, diff_union_of_subset hts]"
},
{
"state_after": "no goals",
"state_before": "case hst\nα : Type u_1\nβ : Type ?u.7261\nE : Type u_2\nF : Type ?u.7267\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht : MeasurableSet t\nhfs : IntegrableOn f s\nhts : t ⊆ s\n⊢ Disjoint (s \\ t) t\n\ncase ht\nα : Type u_1\nβ : Type ?u.7261\nE : Type u_2\nF : Type ?u.7267\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht : MeasurableSet t\nhfs : IntegrableOn f s\nhts : t ⊆ s\n⊢ MeasurableSet t\n\ncase hfs\nα : Type u_1\nβ : Type ?u.7261\nE : Type u_2\nF : Type ?u.7267\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht : MeasurableSet t\nhfs : IntegrableOn f s\nhts : t ⊆ s\n⊢ IntegrableOn (fun x => f x) (s \\ t)\n\ncase hft\nα : Type u_1\nβ : Type ?u.7261\nE : Type u_2\nF : Type ?u.7267\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl l' : Filter α\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nht : MeasurableSet t\nhfs : IntegrableOn f s\nhts : t ⊆ s\n⊢ IntegrableOn (fun x => f x) t",
"tactic": "exacts [disjoint_sdiff_self_left, ht, hfs.mono_set (diff_subset _ _), hfs.mono_set hts]"
}
] |
[
113,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
MeasureTheory.Measure.isHaarMeasure_eq_smul_isHaarMeasure
|
[
{
"state_after": "G : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\n⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν",
"state_before": "G : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\n⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν",
"tactic": "have K : PositiveCompacts G := Classical.arbitrary _"
},
{
"state_after": "G : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\n⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν",
"state_before": "G : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\n⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν",
"tactic": "have νpos : 0 < ν K := measure_pos_of_nonempty_interior _ K.interior_nonempty"
},
{
"state_after": "G : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\nνne : ↑↑ν ↑K ≠ ⊤\n⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν",
"state_before": "G : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\n⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν",
"tactic": "have νne : ν K ≠ ∞ := K.isCompact.measure_lt_top.ne"
},
{
"state_after": "case refine'_1\nG : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\nνne : ↑↑ν ↑K ≠ ⊤\n⊢ ↑↑μ ↑K / ↑↑ν ↑K ≠ 0\n\ncase refine'_2\nG : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\nνne : ↑↑ν ↑K ≠ ⊤\n⊢ ↑↑μ ↑K / ↑↑ν ↑K ≠ ⊤\n\ncase refine'_3\nG : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\nνne : ↑↑ν ↑K ≠ ⊤\n⊢ μ = (↑↑μ ↑K / ↑↑ν ↑K) • ν",
"state_before": "G : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\nνne : ↑↑ν ↑K ≠ ⊤\n⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν",
"tactic": "refine' ⟨μ K / ν K, _, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nG : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\nνne : ↑↑ν ↑K ≠ ⊤\n⊢ ↑↑μ ↑K / ↑↑ν ↑K ≠ 0",
"tactic": "simp only [νne, (μ.measure_pos_of_nonempty_interior K.interior_nonempty).ne', Ne.def,\n ENNReal.div_eq_zero_iff, not_false_iff, or_self_iff]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nG : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\nνne : ↑↑ν ↑K ≠ ⊤\n⊢ ↑↑μ ↑K / ↑↑ν ↑K ≠ ⊤",
"tactic": "simp only [div_eq_mul_inv, νpos.ne', (K.isCompact.measure_lt_top (μ := μ)).ne, or_self_iff,\n ENNReal.inv_eq_top, ENNReal.mul_eq_top, Ne.def, not_false_iff, and_false_iff,\n false_and_iff]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nG : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\nνne : ↑↑ν ↑K ≠ ⊤\n⊢ μ = (↑↑μ ↑K / ↑↑ν ↑K) • ν",
"tactic": "calc\n μ = μ K • haarMeasure K := haarMeasure_unique μ K\n _ = (μ K / ν K) • ν K • haarMeasure K := by\n rw [smul_smul, div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel νpos.ne' νne, mul_one]\n _ = (μ K / ν K) • ν := by rw [← haarMeasure_unique ν K]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\nνne : ↑↑ν ↑K ≠ ⊤\n⊢ ↑↑μ ↑K • haarMeasure K = (↑↑μ ↑K / ↑↑ν ↑K) • ↑↑ν ↑K • haarMeasure K",
"tactic": "rw [smul_smul, div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel νpos.ne' νne, mul_one]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : T2Space G\ninst✝⁶ : TopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : SecondCountableTopology G\ninst✝² : LocallyCompactSpace G\nμ ν : Measure G\ninst✝¹ : IsHaarMeasure μ\ninst✝ : IsHaarMeasure ν\nK : PositiveCompacts G\nνpos : 0 < ↑↑ν ↑K\nνne : ↑↑ν ↑K ≠ ⊤\n⊢ (↑↑μ ↑K / ↑↑ν ↑K) • ↑↑ν ↑K • haarMeasure K = (↑↑μ ↑K / ↑↑ν ↑K) • ν",
"tactic": "rw [← haarMeasure_unique ν K]"
}
] |
[
736,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
721,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
AffineIndependent.mono
|
[] |
[
346,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
11
] |
Mathlib/Algebra/Homology/Exact.lean
|
CategoryTheory.Preadditive.exact_of_iso_of_exact'
|
[] |
[
120,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.IsLimit.ne_zero
|
[] |
[
840,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
839,
11
] |
Mathlib/Init/Data/Bool/Lemmas.lean
|
Bool.decide_congr
|
[
{
"state_after": "no goals",
"state_before": "p q : Prop\ninst✝¹ : Decidable p\ninst✝ : Decidable q\nh : p ↔ q\n⊢ decide p = decide q",
"tactic": "cases h' : decide q with\n| false => exact decide_false (mt h.1 <| of_decide_false h')\n| true => exact decide_true (h.2 <| of_decide_true h')"
},
{
"state_after": "no goals",
"state_before": "case false\np q : Prop\ninst✝¹ : Decidable p\ninst✝ : Decidable q\nh : p ↔ q\nh' : decide q = false\n⊢ decide p = false",
"tactic": "exact decide_false (mt h.1 <| of_decide_false h')"
},
{
"state_after": "no goals",
"state_before": "case true\np q : Prop\ninst✝¹ : Decidable p\ninst✝ : Decidable q\nh : p ↔ q\nh' : decide q = true\n⊢ decide p = true",
"tactic": "exact decide_true (h.2 <| of_decide_true h')"
}
] |
[
159,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.mul_apply'
|
[] |
[
1113,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1112,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.mem_toPrepartition
|
[] |
[
57,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Order/Filter/Interval.lean
|
Filter.Tendsto.Ioc
|
[] |
[
87,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
11
] |
Std/Data/Int/DivMod.lean
|
Int.dvd_natAbs
|
[
{
"state_after": "no goals",
"state_before": "a b : Int\ne : b = ↑(natAbs b)\n⊢ a ∣ ↑(natAbs b) ↔ a ∣ b",
"tactic": "rw [← e]"
},
{
"state_after": "no goals",
"state_before": "a b : Int\ne : b = -↑(natAbs b)\n⊢ a ∣ ↑(natAbs b) ↔ a ∣ b",
"tactic": "rw [← Int.dvd_neg, ← e]"
}
] |
[
659,
41
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
656,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.ne_zero
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nh : Irrational x\n⊢ x ≠ 0",
"tactic": "exact_mod_cast h.ne_nat 0"
}
] |
[
175,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.image_biUnion
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝¹ : DecidableEq β\nf✝ g : α → β\ns✝¹ : Finset α\nt✝ : Finset β\na✝ : α\nb c : β\ninst✝ : DecidableEq γ\nf : α → β\ns✝ : Finset α\nt : β → Finset γ\nthis : DecidableEq α\na : α\ns : Finset α\nx✝ : ¬a ∈ s\nih : Finset.biUnion (image f s) t = Finset.biUnion s fun a => t (f a)\n⊢ Finset.biUnion (image f (insert a s)) t = Finset.biUnion (insert a s) fun a => t (f a)",
"tactic": "simp only [image_insert, biUnion_insert, ih]"
}
] |
[
630,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
627,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.pow_of_dvd
|
[
{
"state_after": "M : Type u_1\nN : Type ?u.2172208\nG : Type ?u.2172211\nR : Type ?u.2172214\nS : Type ?u.2172217\nF : Type ?u.2172220\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\np : ℕ\nhp : p ≠ 0\nhdiv : p ∣ k\n⊢ orderOf (ζ ^ p) = k / p",
"state_before": "M : Type u_1\nN : Type ?u.2172208\nG : Type ?u.2172211\nR : Type ?u.2172214\nS : Type ?u.2172217\nF : Type ?u.2172220\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\np : ℕ\nhp : p ≠ 0\nhdiv : p ∣ k\n⊢ IsPrimitiveRoot (ζ ^ p) (k / p)",
"tactic": "suffices orderOf (ζ ^ p) = k / p by exact this ▸ IsPrimitiveRoot.orderOf (ζ ^ p)"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.2172208\nG : Type ?u.2172211\nR : Type ?u.2172214\nS : Type ?u.2172217\nF : Type ?u.2172220\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\np : ℕ\nhp : p ≠ 0\nhdiv : p ∣ k\n⊢ orderOf (ζ ^ p) = k / p",
"tactic": "rw [orderOf_pow' _ hp, ← eq_orderOf h, Nat.gcd_eq_right hdiv]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.2172208\nG : Type ?u.2172211\nR : Type ?u.2172214\nS : Type ?u.2172217\nF : Type ?u.2172220\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : M\nf : F\nh✝ h : IsPrimitiveRoot ζ k\np : ℕ\nhp : p ≠ 0\nhdiv : p ∣ k\nthis : orderOf (ζ ^ p) = k / p\n⊢ IsPrimitiveRoot (ζ ^ p) (k / p)",
"tactic": "exact this ▸ IsPrimitiveRoot.orderOf (ζ ^ p)"
}
] |
[
483,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
480,
1
] |
Mathlib/Order/CompactlyGenerated.lean
|
CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt
|
[
{
"state_after": "ι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\n⊢ sSup s < k",
"state_before": "ι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : IsCompactElement k\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\n⊢ sSup s < k",
"tactic": "rw [isCompactElement_iff_le_of_directed_sSup_le] at hk"
},
{
"state_after": "ι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\nh : ¬sSup s < k\n⊢ False",
"state_before": "ι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\n⊢ sSup s < k",
"tactic": "by_contra h"
},
{
"state_after": "ι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\nh : ¬sSup s < k\nsSup' : sSup s ≤ k\n⊢ False",
"state_before": "ι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\nh : ¬sSup s < k\n⊢ False",
"tactic": "have sSup' : sSup s ≤ k := sSup_le s k fun s hs => (hbelow s hs).le"
},
{
"state_after": "ι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ SupSet.sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\nh : ¬SupSet.sSup s < k\nsSup' : SupSet.sSup s ≤ k\nsSup : SupSet.sSup s = k\n⊢ False",
"state_before": "ι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\nh : ¬sSup s < k\nsSup' : sSup s ≤ k\n⊢ False",
"tactic": "replace sSup : sSup s = k := eq_iff_le_not_lt.mpr ⟨sSup', h⟩"
},
{
"state_after": "case intro.intro\nι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ SupSet.sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\nh : ¬SupSet.sSup s < k\nsSup' : SupSet.sSup s ≤ k\nsSup : SupSet.sSup s = k\nx : α\nhxs : x ∈ s\nhkx : k ≤ x\n⊢ False",
"state_before": "ι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ SupSet.sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\nh : ¬SupSet.sSup s < k\nsSup' : SupSet.sSup s ≤ k\nsSup : SupSet.sSup s = k\n⊢ False",
"tactic": "obtain ⟨x, hxs, hkx⟩ := hk s hemp hdir sSup.symm.le"
},
{
"state_after": "case intro.intro\nι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ SupSet.sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\nh : ¬SupSet.sSup s < k\nsSup' : SupSet.sSup s ≤ k\nsSup : SupSet.sSup s = k\nx : α\nhxs : x ∈ s\nhkx : k ≤ x\nhxk : x < k\n⊢ False",
"state_before": "case intro.intro\nι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ SupSet.sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\nh : ¬SupSet.sSup s < k\nsSup' : SupSet.sSup s ≤ k\nsSup : SupSet.sSup s = k\nx : α\nhxs : x ∈ s\nhkx : k ≤ x\n⊢ False",
"tactic": "obtain hxk := hbelow x hxs"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Sort ?u.12274\nα✝ : Type ?u.12277\ninst✝¹ : CompleteLattice α✝\nf : ι → α✝\nα : Type u_1\ninst✝ : CompleteLattice α\nk : α\nhk : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ SupSet.sSup s → ∃ x, x ∈ s ∧ k ≤ x\ns : Set α\nhemp : Set.Nonempty s\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) s\nhbelow : ∀ (x : α), x ∈ s → x < k\nh : ¬SupSet.sSup s < k\nsSup' : SupSet.sSup s ≤ k\nsSup : SupSet.sSup s = k\nx : α\nhxs : x ∈ s\nhkx : k ≤ x\nhxk : x < k\n⊢ False",
"tactic": "exact hxk.ne (hxk.le.antisymm hkx)"
}
] |
[
184,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigO.const_smul_left
|
[] |
[
1706,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1704,
1
] |
Mathlib/Algebra/Order/Field/Pi.lean
|
Pi.exists_forall_pos_add_lt
|
[
{
"state_after": "case intro\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx y : ι → α\nh : ∀ (i : ι), x i < y i\nval✝ : Fintype ι\n⊢ ∃ ε, 0 < ε ∧ ∀ (i : ι), x i + ε < y i",
"state_before": "α : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx y : ι → α\nh : ∀ (i : ι), x i < y i\n⊢ ∃ ε, 0 < ε ∧ ∀ (i : ι), x i + ε < y i",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "case intro.inl\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx y : ι → α\nh : ∀ (i : ι), x i < y i\nval✝ : Fintype ι\nh✝ : IsEmpty ι\n⊢ ∃ ε, 0 < ε ∧ ∀ (i : ι), x i + ε < y i\n\ncase intro.inr\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx y : ι → α\nh : ∀ (i : ι), x i < y i\nval✝ : Fintype ι\nh✝ : Nonempty ι\n⊢ ∃ ε, 0 < ε ∧ ∀ (i : ι), x i + ε < y i",
"state_before": "case intro\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx y : ι → α\nh : ∀ (i : ι), x i < y i\nval✝ : Fintype ι\n⊢ ∃ ε, 0 < ε ∧ ∀ (i : ι), x i + ε < y i",
"tactic": "cases isEmpty_or_nonempty ι"
},
{
"state_after": "case intro.inr\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx y : ι → α\nh : ∀ (i : ι), x i < y i\nval✝ : Fintype ι\nh✝ : Nonempty ι\nε : ι → α\nhε : ∀ (i : ι), 0 < ε i\nhxε : ∀ (i : ι), x i + ε i = y i\n⊢ ∃ ε, 0 < ε ∧ ∀ (i : ι), x i + ε < y i",
"state_before": "case intro.inr\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx y : ι → α\nh : ∀ (i : ι), x i < y i\nval✝ : Fintype ι\nh✝ : Nonempty ι\n⊢ ∃ ε, 0 < ε ∧ ∀ (i : ι), x i + ε < y i",
"tactic": "choose ε hε hxε using fun i => exists_pos_add_of_lt' (h i)"
},
{
"state_after": "case intro.inr\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx : ι → α\nval✝ : Fintype ι\nh✝ : Nonempty ι\nε : ι → α\nhε : ∀ (i : ι), 0 < ε i\nh : ∀ (i : ι), x i < (x + ε) i\nhxε : ∀ (i : ι), x i + ε i = (x + ε) i\n⊢ ∃ ε_1, 0 < ε_1 ∧ ∀ (i : ι), x i + ε_1 < (x + ε) i",
"state_before": "case intro.inr\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx y : ι → α\nh : ∀ (i : ι), x i < y i\nval✝ : Fintype ι\nh✝ : Nonempty ι\nε : ι → α\nhε : ∀ (i : ι), 0 < ε i\nhxε : ∀ (i : ι), x i + ε i = y i\n⊢ ∃ ε, 0 < ε ∧ ∀ (i : ι), x i + ε < y i",
"tactic": "obtain rfl : x + ε = y := funext hxε"
},
{
"state_after": "case intro.inr\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx : ι → α\nval✝ : Fintype ι\nh✝ : Nonempty ι\nε : ι → α\nhε✝ : ∀ (i : ι), 0 < ε i\nh : ∀ (i : ι), x i < (x + ε) i\nhxε : ∀ (i : ι), x i + ε i = (x + ε) i\nhε : 0 < Finset.inf' Finset.univ (_ : Finset.Nonempty Finset.univ) ε\n⊢ ∃ ε_1, 0 < ε_1 ∧ ∀ (i : ι), x i + ε_1 < (x + ε) i",
"state_before": "case intro.inr\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx : ι → α\nval✝ : Fintype ι\nh✝ : Nonempty ι\nε : ι → α\nhε : ∀ (i : ι), 0 < ε i\nh : ∀ (i : ι), x i < (x + ε) i\nhxε : ∀ (i : ι), x i + ε i = (x + ε) i\n⊢ ∃ ε_1, 0 < ε_1 ∧ ∀ (i : ι), x i + ε_1 < (x + ε) i",
"tactic": "have hε : 0 < Finset.univ.inf' Finset.univ_nonempty ε := (Finset.lt_inf'_iff _).2 fun i _ => hε _"
},
{
"state_after": "no goals",
"state_before": "case intro.inr\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx : ι → α\nval✝ : Fintype ι\nh✝ : Nonempty ι\nε : ι → α\nhε✝ : ∀ (i : ι), 0 < ε i\nh : ∀ (i : ι), x i < (x + ε) i\nhxε : ∀ (i : ι), x i + ε i = (x + ε) i\nhε : 0 < Finset.inf' Finset.univ (_ : Finset.Nonempty Finset.univ) ε\n⊢ ∃ ε_1, 0 < ε_1 ∧ ∀ (i : ι), x i + ε_1 < (x + ε) i",
"tactic": "exact\n ⟨_, half_pos hε, fun i =>\n add_lt_add_left ((half_lt_self hε).trans_le <| Finset.inf'_le _ <| Finset.mem_univ _) _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.inl\nα : Type u_1\nι : Type u_2\ninst✝² : LinearOrderedSemifield α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : Finite ι\nx y : ι → α\nh : ∀ (i : ι), x i < y i\nval✝ : Fintype ι\nh✝ : IsEmpty ι\n⊢ ∃ ε, 0 < ε ∧ ∀ (i : ι), x i + ε < y i",
"tactic": "exact ⟨1, zero_lt_one, isEmptyElim⟩"
}
] |
[
33,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
23,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
Finsupp.sum_univ_single
|
[
{
"state_after": "α : Type u_2\nι : Type ?u.481609\nγ : Type ?u.481612\nA : Type ?u.481615\nB : Type ?u.481618\nC : Type ?u.481621\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf : α → ι →₀ A\ni✝ : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.484751\nM : Type u_1\nM' : Type ?u.484757\nN : Type ?u.484760\nP : Type ?u.484763\nG : Type ?u.484766\nH : Type ?u.484769\nR : Type ?u.484772\nS : Type ?u.484775\ninst✝¹ : AddCommMonoid M\ninst✝ : Fintype α\ni : α\nm : M\n⊢ (if i ∈ univ then m else 0) = m",
"state_before": "α : Type u_2\nι : Type ?u.481609\nγ : Type ?u.481612\nA : Type ?u.481615\nB : Type ?u.481618\nC : Type ?u.481621\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf : α → ι →₀ A\ni✝ : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.484751\nM : Type u_1\nM' : Type ?u.484757\nN : Type ?u.484760\nP : Type ?u.484763\nG : Type ?u.484766\nH : Type ?u.484769\nR : Type ?u.484772\nS : Type ?u.484775\ninst✝¹ : AddCommMonoid M\ninst✝ : Fintype α\ni : α\nm : M\n⊢ ∑ j : α, ↑(single i m) j = m",
"tactic": "classical rw [single, coe_mk, Finset.sum_pi_single']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nι : Type ?u.481609\nγ : Type ?u.481612\nA : Type ?u.481615\nB : Type ?u.481618\nC : Type ?u.481621\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf : α → ι →₀ A\ni✝ : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.484751\nM : Type u_1\nM' : Type ?u.484757\nN : Type ?u.484760\nP : Type ?u.484763\nG : Type ?u.484766\nH : Type ?u.484769\nR : Type ?u.484772\nS : Type ?u.484775\ninst✝¹ : AddCommMonoid M\ninst✝ : Fintype α\ni : α\nm : M\n⊢ (if i ∈ univ then m else 0) = m",
"tactic": "simp"
},
{
"state_after": "α : Type u_2\nι : Type ?u.481609\nγ : Type ?u.481612\nA : Type ?u.481615\nB : Type ?u.481618\nC : Type ?u.481621\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf : α → ι →₀ A\ni✝ : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.484751\nM : Type u_1\nM' : Type ?u.484757\nN : Type ?u.484760\nP : Type ?u.484763\nG : Type ?u.484766\nH : Type ?u.484769\nR : Type ?u.484772\nS : Type ?u.484775\ninst✝¹ : AddCommMonoid M\ninst✝ : Fintype α\ni : α\nm : M\n⊢ (if i ∈ univ then m else 0) = m",
"state_before": "α : Type u_2\nι : Type ?u.481609\nγ : Type ?u.481612\nA : Type ?u.481615\nB : Type ?u.481618\nC : Type ?u.481621\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf : α → ι →₀ A\ni✝ : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.484751\nM : Type u_1\nM' : Type ?u.484757\nN : Type ?u.484760\nP : Type ?u.484763\nG : Type ?u.484766\nH : Type ?u.484769\nR : Type ?u.484772\nS : Type ?u.484775\ninst✝¹ : AddCommMonoid M\ninst✝ : Fintype α\ni : α\nm : M\n⊢ ∑ j : α, ↑(single i m) j = m",
"tactic": "rw [single, coe_mk, Finset.sum_pi_single']"
}
] |
[
473,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
469,
1
] |
Mathlib/GroupTheory/GroupAction/Prod.lean
|
Prod.smul_swap
|
[] |
[
73,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.wellFoundedOn_empty
|
[] |
[
65,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Probability/Integration.lean
|
ProbabilityTheory.IndepFun.integral_mul'
|
[] |
[
301,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
|
AlgebraicTopology.DoldKan.Γ₀.Obj.Termwise.mapMono_δ₀'
|
[
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ' Δ'' : SimplexCategory\ni : Δ' ⟶ Δ\ninst✝ : Mono i\nhi : Isδ₀ i\n⊢ (if h : Δ = Δ' then eqToHom (_ : HomologicalComplex.X K (len Δ) = HomologicalComplex.X K (len Δ'))\n else if h : Isδ₀ i then HomologicalComplex.d K (len Δ) (len Δ') else 0) =\n HomologicalComplex.d K (len Δ) (len Δ')",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ' Δ'' : SimplexCategory\ni : Δ' ⟶ Δ\ninst✝ : Mono i\nhi : Isδ₀ i\n⊢ mapMono K i = HomologicalComplex.d K (len Δ) (len Δ')",
"tactic": "unfold mapMono"
},
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ' Δ'' : SimplexCategory\ni : Δ' ⟶ Δ\ninst✝ : Mono i\nhi : Isδ₀ i\n⊢ Δ ≠ Δ'",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ' Δ'' : SimplexCategory\ni : Δ' ⟶ Δ\ninst✝ : Mono i\nhi : Isδ₀ i\n⊢ (if h : Δ = Δ' then eqToHom (_ : HomologicalComplex.X K (len Δ) = HomologicalComplex.X K (len Δ'))\n else if h : Isδ₀ i then HomologicalComplex.d K (len Δ) (len Δ') else 0) =\n HomologicalComplex.d K (len Δ) (len Δ')",
"tactic": "suffices Δ ≠ Δ' by\n simp only [dif_neg this, dif_pos hi]"
},
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ'' : SimplexCategory\ni : Δ ⟶ Δ\ninst✝ : Mono i\nhi : Isδ₀ i\n⊢ False",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ' Δ'' : SimplexCategory\ni : Δ' ⟶ Δ\ninst✝ : Mono i\nhi : Isδ₀ i\n⊢ Δ ≠ Δ'",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ'' : SimplexCategory\ni : Δ ⟶ Δ\ninst✝ : Mono i\nhi : Isδ₀ i\n⊢ False",
"tactic": "simpa only [self_eq_add_right, Nat.one_ne_zero] using hi.1"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\nK K' : ChainComplex C ℕ\nf : K ⟶ K'\nΔ Δ' Δ'' : SimplexCategory\ni : Δ' ⟶ Δ\ninst✝ : Mono i\nhi : Isδ₀ i\nthis : Δ ≠ Δ'\n⊢ (if h : Δ = Δ' then eqToHom (_ : HomologicalComplex.X K (len Δ) = HomologicalComplex.X K (len Δ'))\n else if h : Isδ₀ i then HomologicalComplex.d K (len Δ) (len Δ') else 0) =\n HomologicalComplex.d K (len Δ) (len Δ')",
"tactic": "simp only [dif_neg this, dif_pos hi]"
}
] |
[
117,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.compMeasurable_mk
|
[] |
[
252,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/CategoryTheory/Category/TwoP.lean
|
TwoP.swapEquiv_symm
|
[] |
[
106,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/Set/Intervals/Pi.lean
|
Set.image_update_uIcc_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Lattice (α i)\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\ni : ι\nb : α i\n⊢ update f i '' uIcc (f i) b = uIcc f (update f i b)",
"tactic": "simpa using image_update_uIcc f i (f i) b"
}
] |
[
306,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.characteristic_iff_map_le
|
[
{
"state_after": "G : Type u_1\nG' : Type ?u.352996\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.353005\ninst✝ : AddGroup A\nH K : Subgroup G\n⊢ Characteristic H ↔ ∀ (ϕ : G ≃* G), comap (MulEquiv.toMonoidHom (MulEquiv.symm ϕ)) H ≤ H",
"state_before": "G : Type u_1\nG' : Type ?u.352996\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.353005\ninst✝ : AddGroup A\nH K : Subgroup G\n⊢ Characteristic H ↔ ∀ (ϕ : G ≃* G), map (MulEquiv.toMonoidHom ϕ) H ≤ H",
"tactic": "simp_rw [map_equiv_eq_comap_symm]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.352996\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.353005\ninst✝ : AddGroup A\nH K : Subgroup G\n⊢ Characteristic H ↔ ∀ (ϕ : G ≃* G), comap (MulEquiv.toMonoidHom (MulEquiv.symm ϕ)) H ≤ H",
"tactic": "exact characteristic_iff_comap_le.trans ⟨fun h ϕ => h ϕ.symm, fun h ϕ => h ϕ.symm⟩"
}
] |
[
2031,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2029,
1
] |
Mathlib/CategoryTheory/Closed/Cartesian.lean
|
CategoryTheory.CartesianClosed.eq_curry_iff
|
[] |
[
220,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
lt_of_mul_lt_mul_left'
|
[] |
[
133,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
finEquivPowers_apply
|
[] |
[
749,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
747,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.toDual_lt_toDual_iff
|
[] |
[
1024,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1023,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.self_add_star'
|
[] |
[
1109,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1108,
8
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biproduct.bicone_ι
|
[] |
[
406,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.norm_lt_iff_of_compact
|
[
{
"state_after": "F : Type ?u.1008073\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : SeminormedAddCommGroup β\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nf : α →ᵇ β\nM : ℝ\nM0 : 0 < M\n⊢ dist f 0 < M ↔ ∀ (x : α), dist (↑f x) 0 < M",
"state_before": "F : Type ?u.1008073\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : SeminormedAddCommGroup β\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nf : α →ᵇ β\nM : ℝ\nM0 : 0 < M\n⊢ ‖f‖ < M ↔ ∀ (x : α), ‖↑f x‖ < M",
"tactic": "simp_rw [norm_def, ← dist_zero_right]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.1008073\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : SeminormedAddCommGroup β\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nf : α →ᵇ β\nM : ℝ\nM0 : 0 < M\n⊢ dist f 0 < M ↔ ∀ (x : α), dist (↑f x) 0 < M",
"tactic": "exact dist_lt_iff_of_compact M0"
}
] |
[
853,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
850,
1
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
GromovHausdorff.candidates_symm
|
[] |
[
124,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
9
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_comp_val_Iio_atBot
|
[
{
"state_after": "ι : Type ?u.340608\nι' : Type ?u.340611\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.340620\ninst✝¹ : SemilatticeInf α\ninst✝ : NoMinOrder α\na : α\nf : α → β\nl : Filter β\n⊢ Tendsto (fun x => f ↑x) atBot l ↔ Tendsto (f ∘ Subtype.val) atBot l",
"state_before": "ι : Type ?u.340608\nι' : Type ?u.340611\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.340620\ninst✝¹ : SemilatticeInf α\ninst✝ : NoMinOrder α\na : α\nf : α → β\nl : Filter β\n⊢ Tendsto (fun x => f ↑x) atBot l ↔ Tendsto f atBot l",
"tactic": "rw [← map_val_Iio_atBot a, tendsto_map'_iff]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.340608\nι' : Type ?u.340611\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.340620\ninst✝¹ : SemilatticeInf α\ninst✝ : NoMinOrder α\na : α\nf : α → β\nl : Filter β\n⊢ Tendsto (fun x => f ↑x) atBot l ↔ Tendsto (f ∘ Subtype.val) atBot l",
"tactic": "rfl"
}
] |
[
1630,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1628,
1
] |
Mathlib/Data/Set/Sigma.lean
|
Set.sigma_preimage_left
|
[] |
[
167,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Topology/DenseEmbedding.lean
|
DenseInducing.closure_image_mem_nhds
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1224\nδ : Type ?u.1227\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ns : Set α\na : α\ndi : DenseInducing i\nhs : ∃ i_1, (i a ∈ i_1 ∧ IsOpen i_1) ∧ i ⁻¹' i_1 ⊆ s\n⊢ closure (i '' s) ∈ 𝓝 (i a)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1224\nδ : Type ?u.1227\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ns : Set α\na : α\ndi : DenseInducing i\nhs : s ∈ 𝓝 a\n⊢ closure (i '' s) ∈ 𝓝 (i a)",
"tactic": "rw [di.nhds_eq_comap a, ((nhds_basis_opens _).comap _).mem_iff] at hs"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1224\nδ : Type ?u.1227\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ns : Set α\na : α\ndi : DenseInducing i\nU : Set β\nsub : i ⁻¹' U ⊆ s\nhaU : i a ∈ U\nhUo : IsOpen U\n⊢ closure (i '' s) ∈ 𝓝 (i a)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1224\nδ : Type ?u.1227\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ns : Set α\na : α\ndi : DenseInducing i\nhs : ∃ i_1, (i a ∈ i_1 ∧ IsOpen i_1) ∧ i ⁻¹' i_1 ⊆ s\n⊢ closure (i '' s) ∈ 𝓝 (i a)",
"tactic": "rcases hs with ⟨U, ⟨haU, hUo⟩, sub : i ⁻¹' U ⊆ s⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1224\nδ : Type ?u.1227\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ns : Set α\na : α\ndi : DenseInducing i\nU : Set β\nsub : i ⁻¹' U ⊆ s\nhaU : i a ∈ U\nhUo : IsOpen U\n⊢ U ⊆ closure (i '' s)",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1224\nδ : Type ?u.1227\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ns : Set α\na : α\ndi : DenseInducing i\nU : Set β\nsub : i ⁻¹' U ⊆ s\nhaU : i a ∈ U\nhUo : IsOpen U\n⊢ closure (i '' s) ∈ 𝓝 (i a)",
"tactic": "refine' mem_of_superset (hUo.mem_nhds haU) _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.1224\nδ : Type ?u.1227\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ns : Set α\na : α\ndi : DenseInducing i\nU : Set β\nsub : i ⁻¹' U ⊆ s\nhaU : i a ∈ U\nhUo : IsOpen U\n⊢ U ⊆ closure (i '' s)",
"tactic": "calc\n U ⊆ closure (i '' (i ⁻¹' U)) := di.dense.subset_closure_image_preimage_of_isOpen hUo\n _ ⊆ closure (i '' s) := closure_mono (image_subset i sub)"
}
] |
[
77,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
TopologicalGroup.t2Space
|
[
{
"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✝ : T0Space G\nthis : T3Space G\n⊢ T2Space G",
"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✝ : T0Space G\n⊢ T2Space G",
"tactic": "haveI := TopologicalGroup.t3Space G"
},
{
"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✝ : T0Space G\nthis : T3Space G\n⊢ T2Space G",
"tactic": "infer_instance"
}
] |
[
1454,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1452,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Linear.lean
|
IsBoundedLinearMap.hasFDerivAtFilter
|
[] |
[
112,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
ordConnected_iff_upperClosure_inter_lowerClosure
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.164586\nγ : Type ?u.164589\nι : Sort ?u.164592\nκ : ι → Sort ?u.164597\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns t : Set α\nx : α\nh : ↑(upperClosure s) ∩ ↑(lowerClosure s) = s\n⊢ OrdConnected s",
"state_before": "α : Type u_1\nβ : Type ?u.164586\nγ : Type ?u.164589\nι : Sort ?u.164592\nκ : ι → Sort ?u.164597\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns t : Set α\nx : α\n⊢ OrdConnected s ↔ ↑(upperClosure s) ∩ ↑(lowerClosure s) = s",
"tactic": "refine' ⟨Set.OrdConnected.upperClosure_inter_lowerClosure, fun h => _⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.164586\nγ : Type ?u.164589\nι : Sort ?u.164592\nκ : ι → Sort ?u.164597\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns t : Set α\nx : α\nh : ↑(upperClosure s) ∩ ↑(lowerClosure s) = s\n⊢ OrdConnected (↑(upperClosure s) ∩ ↑(lowerClosure s))",
"state_before": "α : Type u_1\nβ : Type ?u.164586\nγ : Type ?u.164589\nι : Sort ?u.164592\nκ : ι → Sort ?u.164597\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns t : Set α\nx : α\nh : ↑(upperClosure s) ∩ ↑(lowerClosure s) = s\n⊢ OrdConnected s",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.164586\nγ : Type ?u.164589\nι : Sort ?u.164592\nκ : ι → Sort ?u.164597\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns t : Set α\nx : α\nh : ↑(upperClosure s) ∩ ↑(lowerClosure s) = s\n⊢ OrdConnected (↑(upperClosure s) ∩ ↑(lowerClosure s))",
"tactic": "exact (UpperSet.upper _).ordConnected.inter (LowerSet.lower _).ordConnected"
}
] |
[
1472,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1468,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.add_left_cancel
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.54905\nβ : Type ?u.54908\nγ : Type ?u.54911\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b c : Ordinal\n⊢ a + b = a + c ↔ b = c",
"tactic": "simp only [le_antisymm_iff, add_le_add_iff_left]"
}
] |
[
119,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Algebra/Field/Basic.lean
|
inv_sub_inv'
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.42064\nβ : Type ?u.42067\nK : Type u_1\ninst✝ : DivisionRing K\na✝ b✝ c d a b : K\nha : a ≠ 0\nhb : b ≠ 0\n⊢ a⁻¹ - b⁻¹ = a⁻¹ * (b - a) * b⁻¹",
"tactic": "rw [mul_sub, sub_mul, mul_inv_cancel_right₀ hb, inv_mul_cancel ha, one_mul]"
}
] |
[
178,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/CategoryTheory/GlueData.lean
|
CategoryTheory.GlueData.t'_jii
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nC' : Type u₂\ninst✝ : Category C'\nD : GlueData C\ni j : D.J\n⊢ t' D j i i = (t' D j i i ≫ pullback.snd) ≫ inv pullback.snd",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nC' : Type u₂\ninst✝ : Category C'\nD : GlueData C\ni j : D.J\n⊢ t' D j i i = pullback.fst ≫ t D j i ≫ inv pullback.snd",
"tactic": "rw [← Category.assoc, ← D.t_fac]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nC' : Type u₂\ninst✝ : Category C'\nD : GlueData C\ni j : D.J\n⊢ t' D j i i = (t' D j i i ≫ pullback.snd) ≫ inv pullback.snd",
"tactic": "simp"
}
] |
[
93,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Computability/PartrecCode.lean
|
Nat.Partrec.Code.eval_part
|
[
{
"state_after": "no goals",
"state_before": "a : Code × ℕ\n⊢ (rfindOpt fun b =>\n evaln (((a, b).snd, (a, b).fst.fst), (a, b).fst.snd).fst.fst\n (((a, b).snd, (a, b).fst.fst), (a, b).fst.snd).fst.snd (((a, b).snd, (a, b).fst.fst), (a, b).fst.snd).snd) =\n eval a.fst a.snd",
"tactic": "simp [eval_eq_rfindOpt]"
}
] |
[
1164,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1161,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.divInt_self
|
[
{
"state_after": "no goals",
"state_before": "a : Rat\n⊢ a.num /. ↑a.den = a",
"tactic": "rw [divInt_ofNat, mkRat_self]"
}
] |
[
122,
87
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
122,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.comp_affineMap
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nα : Type ?u.743875\nβ : Type u_4\nι : Type ?u.743881\ninst✝⁶ : LinearOrderedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : AddCommGroup F\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 F\ninst✝ : SMul 𝕜 β\nf : F → β\ng : E →ᵃ[𝕜] F\ns : Set F\nhf : ConvexOn 𝕜 s f\nx : E\nhx : x ∈ ↑g ⁻¹' s\ny : E\nhy : y ∈ ↑g ⁻¹' s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f (↑g (a • x + b • y)) = f (a • ↑g x + b • ↑g y)",
"tactic": "rw [Convex.combo_affine_apply hab]"
}
] |
[
1016,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1010,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Functor.rightAdjoint_of_isEquivalence
|
[] |
[
659,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/RingTheory/Polynomial/Dickson.lean
|
Polynomial.C_half_mul_two_eq_one
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.151322\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nk : ℕ\na : R\ninst✝ : Invertible 2\n⊢ ↑C ⅟2 * 2 = 1",
"tactic": "rw [mul_comm, two_mul_C_half_eq_one]"
}
] |
[
154,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
9
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
NonarchAddGroupNorm.coe_sup
|
[] |
[
958,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
957,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.univ_pi_update
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_3\nβ✝ : ι → Type ?u.152906\ns s₁ s₂ : Set ι\nt✝ t₁ t₂ : (i : ι) → Set (α i)\ni✝ : ι\ninst✝ : DecidableEq ι\nβ : ι → Type u_2\ni : ι\nf : (j : ι) → α j\na : α i\nt : (j : ι) → α j → Set (β j)\n⊢ (pi univ fun j => t j (update f i a j)) = {x | x i ∈ t i a} ∩ pi ({i}ᶜ) fun j => t j (f j)",
"tactic": "rw [compl_eq_univ_diff, ← pi_update_of_mem (mem_univ _)]"
}
] |
[
790,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
787,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux
|
[
{
"state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\n⊢ P ((g ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑(LocalHomeomorph.symm (LocalHomeomorph.symm e ≫ₕ e')))\n (↑(LocalHomeomorph.symm (LocalHomeomorph.symm e ≫ₕ e')) ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s))\n (↑(LocalHomeomorph.symm e ≫ₕ e') (↑e x)) ↔\n P (g ∘ ↑(LocalHomeomorph.symm e')) (↑(LocalHomeomorph.symm e') ⁻¹' s) (↑e' x)\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\n⊢ ↑e x ∈ (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\n⊢ P (g ∘ ↑(LocalHomeomorph.symm e)) (↑(LocalHomeomorph.symm e) ⁻¹' s) (↑e x) ↔\n P (g ∘ ↑(LocalHomeomorph.symm e')) (↑(LocalHomeomorph.symm e') ⁻¹' s) (↑e' x)",
"tactic": "rw [← hG.right_invariance (compatible_of_mem_maximalAtlas he he')]"
},
{
"state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\n⊢ ↑e x ∈ (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\n⊢ P ((g ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑(LocalHomeomorph.symm (LocalHomeomorph.symm e ≫ₕ e')))\n (↑(LocalHomeomorph.symm (LocalHomeomorph.symm e ≫ₕ e')) ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s))\n (↑(LocalHomeomorph.symm e ≫ₕ e') (↑e x)) ↔\n P (g ∘ ↑(LocalHomeomorph.symm e')) (↑(LocalHomeomorph.symm e') ⁻¹' s) (↑e' x)",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\n⊢ P ((g ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑(LocalHomeomorph.symm (LocalHomeomorph.symm e ≫ₕ e')))\n (↑(LocalHomeomorph.symm (LocalHomeomorph.symm e ≫ₕ e')) ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s))\n (↑(LocalHomeomorph.symm e ≫ₕ e') (↑e x)) ↔\n P (g ∘ ↑(LocalHomeomorph.symm e')) (↑(LocalHomeomorph.symm e') ⁻¹' s) (↑e' x)\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\n⊢ ↑e x ∈ (LocalHomeomorph.symm e ≫ₕ e').toLocalEquiv.source",
"tactic": "swap"
},
{
"state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\n⊢ P ((g ∘ ↑(LocalHomeomorph.symm e)) ∘ ↑(LocalHomeomorph.symm (LocalHomeomorph.symm e ≫ₕ e')))\n (↑(LocalHomeomorph.symm (LocalHomeomorph.symm e ≫ₕ e')) ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s)) (↑e' x) ↔\n P (g ∘ ↑(LocalHomeomorph.symm e')) (↑(LocalHomeomorph.symm e') ⁻¹' s) (↑e' x)",
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"state_after": "case refine'_1\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\n⊢ ↑(LocalHomeomorph.symm e') ⁻¹' e.source ∈ 𝓝 (↑e' x)\n\ncase refine'_2\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\ny : H\nhy : y ∈ ↑(LocalHomeomorph.symm e') ⁻¹' e.source\n⊢ y ∈ ↑(LocalHomeomorph.symm (LocalHomeomorph.symm e ≫ₕ e')) ⁻¹' (↑(LocalHomeomorph.symm e) ⁻¹' s) ↔\n y ∈ ↑(LocalHomeomorph.symm e') ⁻¹' s",
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"tactic": "refine' (eventually_of_mem _ fun y (hy : y ∈ e'.symm ⁻¹' e.source) ↦ _).set_eq"
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{
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"tactic": "refine' (e'.symm.continuousAt <| e'.mapsTo xe').preimage_mem_nhds (e.open_source.mem_nhds _)"
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"tactic": "refine' ((e'.eventually_nhds' _ xe').mpr <| e.eventually_left_inverse xe).mono fun y hy ↦ _"
},
{
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{
"state_after": "no goals",
"state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type ?u.16176\nX : Type ?u.16179\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\ninst✝¹ : ChartedSpace H' M'\ninst✝ : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng✝ g' : M → M'\ns t : Set M\nx : M\nQ : (H → H) → Set H → H → Prop\nhG : LocalInvariantProp G G' P\ng : M → H'\nhe : e ∈ maximalAtlas M G\nxe : x ∈ e.source\nhe' : e' ∈ maximalAtlas M G\nxe' : x ∈ e'.source\ny : H\nhy : ↑(LocalHomeomorph.symm e) (↑e (↑(LocalHomeomorph.symm e') y)) = ↑(LocalHomeomorph.symm e') y\n⊢ g (↑(LocalHomeomorph.symm e) (↑e (↑(LocalHomeomorph.symm e') y))) = g (↑(LocalHomeomorph.symm e') y)",
"tactic": "rw [hy]"
}
] |
[
271,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.natCast_ne_top
|
[] |
[
358,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.coprod.pentagon
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nX✝ Y✝ : C\ninst✝¹ : Category C\ninst✝ : HasBinaryCoproducts C\nW X Y Z : C\n⊢ map (associator W X Y).hom (𝟙 Z) ≫ (associator W (X ⨿ Y) Z).hom ≫ map (𝟙 W) (associator X Y Z).hom =\n (associator (W ⨿ X) Y Z).hom ≫ (associator W X (Y ⨿ Z)).hom",
"tactic": "simp"
}
] |
[
1148,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1144,
1
] |
Mathlib/Data/List/Perm.lean
|
List.permutations'Aux_nthLe_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ s : List α\nx : α\n⊢ 0 < length (permutations'Aux x s)",
"tactic": "simp"
}
] |
[
1390,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1387,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.sup_le_lsub
|
[] |
[
1609,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1608,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.land_bit
|
[
{
"state_after": "no goals",
"state_before": "a : Bool\nm : ℤ\nb : Bool\nn : ℤ\n⊢ land (bit a m) (bit b n) = bit (a && b) (land m n)",
"tactic": "rw [← bitwise_and, bitwise_bit]"
}
] |
[
315,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
singleton_mul_mem_nhds
|
[
{
"state_after": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul α α\ns t : Set α\na b : α\nh : s ∈ 𝓝 b\nthis : a • s ∈ 𝓝 (a • b)\n⊢ {a} * s ∈ 𝓝 (a * b)",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul α α\ns t : Set α\na b : α\nh : s ∈ 𝓝 b\n⊢ {a} * s ∈ 𝓝 (a * b)",
"tactic": "have := smul_mem_nhds a h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace α\ninst✝¹ : Group α\ninst✝ : ContinuousConstSMul α α\ns t : Set α\na b : α\nh : s ∈ 𝓝 b\nthis : a • s ∈ 𝓝 (a • b)\n⊢ {a} * s ∈ 𝓝 (a * b)",
"tactic": "rwa [← singleton_smul] at this"
}
] |
[
1280,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1278,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
CharP.ringChar_ne_zero_of_finite
|
[] |
[
487,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/FieldTheory/IsAlgClosed/Basic.lean
|
IsAlgClosed.exists_eval₂_eq_zero
|
[] |
[
118,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Algebra/Lie/Solvable.lean
|
LieAlgebra.derivedSeries_of_derivedLength_succ
|
[
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\n⊢ derivedLengthOfIdeal R L I = k + 1 ↔ derivedSeriesOfIdeal R L (k + 1) I = ⊥ ∧ derivedSeriesOfIdeal R L k I ≠ ⊥",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\n⊢ derivedLengthOfIdeal R L I = k + 1 ↔\n IsLieAbelian { x // x ∈ ↑(derivedSeriesOfIdeal R L k I) } ∧ derivedSeriesOfIdeal R L k I ≠ ⊥",
"tactic": "rw [abelian_iff_derived_succ_eq_bot]"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\n⊢ derivedLengthOfIdeal R L I = k + 1 ↔ derivedSeriesOfIdeal R L (k + 1) I = ⊥ ∧ derivedSeriesOfIdeal R L k I ≠ ⊥",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\n⊢ derivedLengthOfIdeal R L I = k + 1 ↔ derivedSeriesOfIdeal R L (k + 1) I = ⊥ ∧ derivedSeriesOfIdeal R L k I ≠ ⊥",
"tactic": "let s := { k | derivedSeriesOfIdeal R L k I = ⊥ }"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\n⊢ sInf s = k + 1 ↔ k + 1 ∈ s ∧ ¬k ∈ s",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\n⊢ derivedLengthOfIdeal R L I = k + 1 ↔ derivedSeriesOfIdeal R L (k + 1) I = ⊥ ∧ derivedSeriesOfIdeal R L k I ≠ ⊥",
"tactic": "change sInf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nhs : ∀ (k₁ k₂ : ℕ), k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s\n⊢ sInf s = k + 1 ↔ k + 1 ∈ s ∧ ¬k ∈ s",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\n⊢ sInf s = k + 1 ↔ k + 1 ∈ s ∧ ¬k ∈ s",
"tactic": "have hs : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s := by\n intro k₁ k₂ h₁₂ h₁\n suffices derivedSeriesOfIdeal R L k₂ I ≤ ⊥ by exact eq_bot_iff.mpr this\n change derivedSeriesOfIdeal R L k₁ I = ⊥ at h₁ ; rw [← h₁]\n exact derivedSeriesOfIdeal_antitone I h₁₂"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nhs : ∀ (k₁ k₂ : ℕ), k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s\n⊢ sInf s = k + 1 ↔ k + 1 ∈ s ∧ ¬k ∈ s",
"tactic": "exact Nat.sInf_upward_closed_eq_succ_iff hs k"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : k₁ ∈ s\n⊢ k₂ ∈ s",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\n⊢ ∀ (k₁ k₂ : ℕ), k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s",
"tactic": "intro k₁ k₂ h₁₂ h₁"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : k₁ ∈ s\n⊢ derivedSeriesOfIdeal R L k₂ I ≤ ⊥",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : k₁ ∈ s\n⊢ k₂ ∈ s",
"tactic": "suffices derivedSeriesOfIdeal R L k₂ I ≤ ⊥ by exact eq_bot_iff.mpr this"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : derivedSeriesOfIdeal R L k₁ I = ⊥\n⊢ derivedSeriesOfIdeal R L k₂ I ≤ ⊥",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : k₁ ∈ s\n⊢ derivedSeriesOfIdeal R L k₂ I ≤ ⊥",
"tactic": "change derivedSeriesOfIdeal R L k₁ I = ⊥ at h₁"
},
{
"state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : derivedSeriesOfIdeal R L k₁ I = ⊥\n⊢ derivedSeriesOfIdeal R L k₂ I ≤ derivedSeriesOfIdeal R L k₁ I",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : derivedSeriesOfIdeal R L k₁ I = ⊥\n⊢ derivedSeriesOfIdeal R L k₂ I ≤ ⊥",
"tactic": "rw [← h₁]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : derivedSeriesOfIdeal R L k₁ I = ⊥\n⊢ derivedSeriesOfIdeal R L k₂ I ≤ derivedSeriesOfIdeal R L k₁ I",
"tactic": "exact derivedSeriesOfIdeal_antitone I h₁₂"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nI✝ J : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\nI : LieIdeal R L\nk : ℕ\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk₁ k₂ : ℕ\nh₁₂ : k₁ ≤ k₂\nh₁ : k₁ ∈ s\nthis : derivedSeriesOfIdeal R L k₂ I ≤ ⊥\n⊢ k₂ ∈ s",
"tactic": "exact eq_bot_iff.mpr this"
}
] |
[
323,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.log_pos
|
[
{
"state_after": "no goals",
"state_before": "b o : Ordinal\nhb : 1 < b\nho : o ≠ 0\nhbo : b ≤ o\n⊢ 0 < log b o",
"tactic": "rwa [← succ_le_iff, succ_zero, ← opow_le_iff_le_log hb ho, opow_one]"
}
] |
[
344,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.distortion_biUnionPrepartition
|
[] |
[
442,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/CategoryTheory/Adhesive.lean
|
CategoryTheory.IsPushout.isVanKampen_iff
|
[
{
"state_after": "case mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\n⊢ IsVanKampen H → IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\n\ncase mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\n⊢ IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)) → IsVanKampen H",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\n⊢ IsVanKampen H ↔ IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))",
"tactic": "constructor"
},
{
"state_after": "case mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ Nonempty (IsColimit c') ↔\n ∀ (j : WalkingSpan), IsPullback (c'.ι.app j) (α.app j) fα ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)",
"state_before": "case mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\n⊢ IsVanKampen H → IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))",
"tactic": "intro H F' c' α fα eα hα"
},
{
"state_after": "case mp.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ Nonempty (IsColimit c') ↔\n IsPushout (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app WalkingSpan.left)\n (c'.ι.app WalkingSpan.right)\n\ncase mp.refine'_2\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ CommSq (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h\n\ncase mp.refine'_3\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ CommSq (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i\n\ncase mp.refine'_4\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ CommSq (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app WalkingSpan.left)\n (c'.ι.app WalkingSpan.right)\n\ncase mp.refine'_5\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h ∧\n IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i ↔\n ∀ (j : WalkingSpan), IsPullback (c'.ι.app j) (α.app j) fα ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)",
"state_before": "case mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ Nonempty (IsColimit c') ↔\n ∀ (j : WalkingSpan), IsPullback (c'.ι.app j) (α.app j) fα ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)",
"tactic": "refine' Iff.trans _\n ((H (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app _) (c'.ι.app _)\n (α.app _) (α.app _) (α.app _) fα (by convert hα WalkingSpan.Hom.fst)\n (by convert hα WalkingSpan.Hom.snd) _ _ _).trans _)"
},
{
"state_after": "case mp.refine'_5.mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h ∧\n IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i →\n ∀ (j : WalkingSpan), IsPullback (c'.ι.app j) (α.app j) fα ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)\n\ncase mp.refine'_5.mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ (∀ (j : WalkingSpan), IsPullback (c'.ι.app j) (α.app j) fα ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)) →\n IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h ∧\n IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i",
"state_before": "case mp.refine'_5\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h ∧\n IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i ↔\n ∀ (j : WalkingSpan), IsPullback (c'.ι.app j) (α.app j) fα ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ IsPullback (F'.map WalkingSpan.Hom.fst) (α.app WalkingSpan.zero) (α.app WalkingSpan.left) f",
"tactic": "convert hα WalkingSpan.Hom.fst"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ IsPullback (F'.map WalkingSpan.Hom.snd) (α.app WalkingSpan.zero) (α.app WalkingSpan.right) g",
"tactic": "convert hα WalkingSpan.Hom.snd"
},
{
"state_after": "case mp.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nthis : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right\n⊢ Nonempty (IsColimit c') ↔\n IsPushout (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app WalkingSpan.left)\n (c'.ι.app WalkingSpan.right)",
"state_before": "case mp.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ Nonempty (IsColimit c') ↔\n IsPushout (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app WalkingSpan.left)\n (c'.ι.app WalkingSpan.right)",
"tactic": "have : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left =\n F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right := by\n simp only [Cocone.w]"
},
{
"state_after": "case mp.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nthis : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right\n⊢ Nonempty (IsColimit (PushoutCocone.mk (c'.ι.app WalkingSpan.left) (c'.ι.app WalkingSpan.right) this)) ↔\n IsPushout (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app WalkingSpan.left)\n (c'.ι.app WalkingSpan.right)\n\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nthis : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right\n⊢ (Cocones.precompose (diagramIsoSpan F').inv).obj c' ≅\n PushoutCocone.mk (c'.ι.app WalkingSpan.left) (c'.ι.app WalkingSpan.right) this",
"state_before": "case mp.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nthis : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right\n⊢ Nonempty (IsColimit c') ↔\n IsPushout (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app WalkingSpan.left)\n (c'.ι.app WalkingSpan.right)",
"tactic": "rw [(IsColimit.equivOfNatIsoOfIso (diagramIsoSpan F') c' (PushoutCocone.mk _ _ this)\n _).nonempty_congr]"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right",
"tactic": "simp only [Cocone.w]"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nthis : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right\n⊢ Nonempty (IsColimit (PushoutCocone.mk (c'.ι.app WalkingSpan.left) (c'.ι.app WalkingSpan.right) this)) ↔\n IsPushout (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app WalkingSpan.left)\n (c'.ι.app WalkingSpan.right)",
"tactic": "exact ⟨fun h => ⟨⟨this⟩, h⟩, fun h => h.2⟩"
},
{
"state_after": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nthis : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right\n⊢ ∀ (j : WalkingSpan),\n ((Cocones.precompose (diagramIsoSpan F').inv).obj c').ι.app j ≫ (Iso.refl c'.pt).hom =\n (PushoutCocone.mk (c'.ι.app WalkingSpan.left) (c'.ι.app WalkingSpan.right) this).ι.app j",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nthis : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right\n⊢ (Cocones.precompose (diagramIsoSpan F').inv).obj c' ≅\n PushoutCocone.mk (c'.ι.app WalkingSpan.left) (c'.ι.app WalkingSpan.right) this",
"tactic": "refine' Cocones.ext (Iso.refl c'.pt) _"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nthis : F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right\n⊢ ∀ (j : WalkingSpan),\n ((Cocones.precompose (diagramIsoSpan F').inv).obj c').ι.app j ≫ (Iso.refl c'.pt).hom =\n (PushoutCocone.mk (c'.ι.app WalkingSpan.left) (c'.ι.app WalkingSpan.right) this).ι.app j",
"tactic": "rintro (_ | _ | _) <;> dsimp <;>\n simp only [c'.w, Category.assoc, Category.id_comp, Category.comp_id]"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_2\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ CommSq (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h",
"tactic": "exact ⟨NatTrans.congr_app eα.symm _⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_3\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ CommSq (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i",
"tactic": "exact ⟨NatTrans.congr_app eα.symm _⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_4\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ CommSq (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app WalkingSpan.left)\n (c'.ι.app WalkingSpan.right)",
"tactic": "exact ⟨by simp⟩"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left = F'.map WalkingSpan.Hom.snd ≫ c'.ι.app WalkingSpan.right",
"tactic": "simp"
},
{
"state_after": "case mp.refine'_5.mp.intro.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nh₁ : IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h\nh₂ : IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i\n⊢ IsPullback (c'.ι.app none) (α.app none) fα ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app none)\n\ncase mp.refine'_5.mp.intro.some.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nh₁ : IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h\nh₂ : IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i\n⊢ IsPullback (c'.ι.app (some WalkingPair.left)) (α.app (some WalkingPair.left)) fα\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app (some WalkingPair.left))\n\ncase mp.refine'_5.mp.intro.some.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nh₁ : IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h\nh₂ : IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i\n⊢ IsPullback (c'.ι.app (some WalkingPair.right)) (α.app (some WalkingPair.right)) fα\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app (some WalkingPair.right))",
"state_before": "case mp.refine'_5.mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h ∧\n IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i →\n ∀ (j : WalkingSpan), IsPullback (c'.ι.app j) (α.app j) fα ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)",
"tactic": "rintro ⟨h₁, h₂⟩ (_ | _ | _)"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_5.mp.intro.some.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nh₁ : IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h\nh₂ : IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i\n⊢ IsPullback (c'.ι.app (some WalkingPair.left)) (α.app (some WalkingPair.left)) fα\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app (some WalkingPair.left))\n\ncase mp.refine'_5.mp.intro.some.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nh₁ : IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h\nh₂ : IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i\n⊢ IsPullback (c'.ι.app (some WalkingPair.right)) (α.app (some WalkingPair.right)) fα\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app (some WalkingPair.right))",
"tactic": "exacts [h₁, h₂]"
},
{
"state_after": "case mp.refine'_5.mp.intro.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nh₁ : IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h\nh₂ : IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i\n⊢ IsPullback (F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left) (α.app none) fα\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app none)",
"state_before": "case mp.refine'_5.mp.intro.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nh₁ : IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h\nh₂ : IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i\n⊢ IsPullback (c'.ι.app none) (α.app none) fα ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app none)",
"tactic": "rw [← c'.w WalkingSpan.Hom.fst]"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_5.mp.intro.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nh₁ : IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h\nh₂ : IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i\n⊢ IsPullback (F'.map WalkingSpan.Hom.fst ≫ c'.ι.app WalkingSpan.left) (α.app none) fα\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app none)",
"tactic": "exact (hα WalkingSpan.Hom.fst).paste_horiz h₁"
},
{
"state_after": "case mp.refine'_5.mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh✝ : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h✝ i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h✝ i (_ : f ≫ h✝ = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h✝ i (_ : f ≫ h✝ = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nh : ∀ (j : WalkingSpan), IsPullback (c'.ι.app j) (α.app j) fα ((PushoutCocone.mk h✝ i (_ : f ≫ h✝ = g ≫ i)).ι.app j)\n⊢ IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h✝ ∧\n IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i",
"state_before": "case mp.refine'_5.mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\n⊢ (∀ (j : WalkingSpan), IsPullback (c'.ι.app j) (α.app j) fα ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)) →\n IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h ∧\n IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_5.mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh✝ : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h✝ i\nH : IsVanKampen H✝\nF' : WalkingSpan ⥤ C\nc' : Cocone F'\nα : F' ⟶ span f g\nfα : c'.pt ⟶ (PushoutCocone.mk h✝ i (_ : f ≫ h✝ = g ≫ i)).pt\neα : α ≫ (PushoutCocone.mk h✝ i (_ : f ≫ h✝ = g ≫ i)).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα\nhα : NatTrans.Equifibered α\nh : ∀ (j : WalkingSpan), IsPullback (c'.ι.app j) (α.app j) fα ((PushoutCocone.mk h✝ i (_ : f ≫ h✝ = g ≫ i)).ι.app j)\n⊢ IsPullback (c'.ι.app WalkingSpan.left) (α.app WalkingSpan.left) fα h✝ ∧\n IsPullback (c'.ι.app WalkingSpan.right) (α.app WalkingSpan.right) fα i",
"tactic": "exact ⟨h _, h _⟩"
},
{
"state_after": "case mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPushout f' g' h' i' ↔ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i",
"state_before": "case mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\n⊢ IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)) → IsVanKampen H",
"tactic": "introv H W' hf hg hh hi w"
},
{
"state_after": "case mpr.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPushout f' g' h' i' ↔ Nonempty (IsColimit (CommSq.cocone w))\n\ncase mpr.refine'_2\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ∀ ⦃X_1 Y_1 : WalkingSpan⦄ (f_1 : X_1 ⟶ Y_1),\n ((span f' g').map f_1 ≫ Option.casesOn Y_1 αW fun val => WalkingPair.casesOn val αX αY) =\n (Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫ (span f g).map f_1\n\ncase mpr.refine'_3\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫\n (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι =\n (CommSq.cocone w).ι ≫ (Functor.const WalkingSpan).map αZ\n\ncase mpr.refine'_4\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ NatTrans.Equifibered (NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY)\n\ncase mpr.refine'_5\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (∀ (j : WalkingSpan),\n IsPullback ((CommSq.cocone w).ι.app j)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app j) αZ\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)) ↔\n IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i",
"state_before": "case mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPushout f' g' h' i' ↔ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i",
"tactic": "refine'\n Iff.trans _ ((H w.cocone ⟨by rintro (_ | _ | _); exacts [αW, αX, αY], _⟩ αZ _ _).trans _)"
},
{
"state_after": "case mpr.refine'_2\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ∀ ⦃X_1 Y_1 : WalkingSpan⦄ (f_1 : X_1 ⟶ Y_1),\n ((span f' g').map f_1 ≫ Option.casesOn Y_1 αW fun val => WalkingPair.casesOn val αX αY) =\n (Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫ (span f g).map f_1\n\ncase mpr.refine'_3\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫\n (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι =\n (CommSq.cocone w).ι ≫ (Functor.const WalkingSpan).map αZ\n\ncase mpr.refine'_4\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ NatTrans.Equifibered (NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY)\n\ncase mpr.refine'_5\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (∀ (j : WalkingSpan),\n IsPullback ((CommSq.cocone w).ι.app j)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app j) αZ\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)) ↔\n IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i\n\ncase mpr.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPushout f' g' h' i' ↔ Nonempty (IsColimit (CommSq.cocone w))",
"state_before": "case mpr.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPushout f' g' h' i' ↔ Nonempty (IsColimit (CommSq.cocone w))\n\ncase mpr.refine'_2\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ∀ ⦃X_1 Y_1 : WalkingSpan⦄ (f_1 : X_1 ⟶ Y_1),\n ((span f' g').map f_1 ≫ Option.casesOn Y_1 αW fun val => WalkingPair.casesOn val αX αY) =\n (Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫ (span f g).map f_1\n\ncase mpr.refine'_3\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫\n (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι =\n (CommSq.cocone w).ι ≫ (Functor.const WalkingSpan).map αZ\n\ncase mpr.refine'_4\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ NatTrans.Equifibered (NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY)\n\ncase mpr.refine'_5\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (∀ (j : WalkingSpan),\n IsPullback ((CommSq.cocone w).ι.app j)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app j) αZ\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)) ↔\n IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i",
"tactic": "rotate_left"
},
{
"state_after": "case none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (span f' g').obj none ⟶ (span f g).obj none\n\ncase some.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (span f' g').obj (some WalkingPair.left) ⟶ (span f g).obj (some WalkingPair.left)\n\ncase some.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (span f' g').obj (some WalkingPair.right) ⟶ (span f g).obj (some WalkingPair.right)",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (X_1 : WalkingSpan) → (span f' g').obj X_1 ⟶ (span f g).obj X_1",
"tactic": "rintro (_ | _ | _)"
},
{
"state_after": "no goals",
"state_before": "case none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (span f' g').obj none ⟶ (span f g).obj none\n\ncase some.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (span f' g').obj (some WalkingPair.left) ⟶ (span f g).obj (some WalkingPair.left)\n\ncase some.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (span f' g').obj (some WalkingPair.right) ⟶ (span f g).obj (some WalkingPair.right)",
"tactic": "exacts [αW, αX, αY]"
},
{
"state_after": "case mpr.refine'_2.id\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni✝ : Y ⟶ Z\nH✝ : IsPushout f g h i✝\nH : IsVanKampenColimit (PushoutCocone.mk h i✝ (_ : f ≫ h = g ≫ i✝))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i✝\nw : CommSq f' g' h' i'\ni : WalkingSpan\n⊢ ((span f' g').map (WidePushoutShape.Hom.id i) ≫ Option.casesOn i αW fun val => WalkingPair.casesOn val αX αY) =\n (Option.casesOn i αW fun val => WalkingPair.casesOn val αX αY) ≫ (span f g).map (WidePushoutShape.Hom.id i)\n\ncase mpr.refine'_2.init.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ((span f' g').map (WidePushoutShape.Hom.init WalkingPair.left) ≫\n Option.casesOn (some WalkingPair.left) αW fun val => WalkingPair.casesOn val αX αY) =\n (Option.casesOn none αW fun val => WalkingPair.casesOn val αX αY) ≫\n (span f g).map (WidePushoutShape.Hom.init WalkingPair.left)\n\ncase mpr.refine'_2.init.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ((span f' g').map (WidePushoutShape.Hom.init WalkingPair.right) ≫\n Option.casesOn (some WalkingPair.right) αW fun val => WalkingPair.casesOn val αX αY) =\n (Option.casesOn none αW fun val => WalkingPair.casesOn val αX αY) ≫\n (span f g).map (WidePushoutShape.Hom.init WalkingPair.right)",
"state_before": "case mpr.refine'_2\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ∀ ⦃X_1 Y_1 : WalkingSpan⦄ (f_1 : X_1 ⟶ Y_1),\n ((span f' g').map f_1 ≫ Option.casesOn Y_1 αW fun val => WalkingPair.casesOn val αX αY) =\n (Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫ (span f g).map f_1",
"tactic": "rintro i _ (_ | _ | _)"
},
{
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"tactic": "exacts [hf.w, hg.w]"
},
{
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"tactic": "dsimp"
},
{
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"tactic": "simp only [Functor.map_id, Category.comp_id, Category.id_comp]"
},
{
"state_after": "case mpr.refine'_3.w.h.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫\n (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι).app\n none =\n ((CommSq.cocone w).ι ≫ (Functor.const WalkingSpan).map αZ).app none\n\ncase mpr.refine'_3.w.h.some.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫\n (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι).app\n (some WalkingPair.left) =\n ((CommSq.cocone w).ι ≫ (Functor.const WalkingSpan).map αZ).app (some WalkingPair.left)\n\ncase mpr.refine'_3.w.h.some.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫\n (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι).app\n (some WalkingPair.right) =\n ((CommSq.cocone w).ι ≫ (Functor.const WalkingSpan).map αZ).app (some WalkingPair.right)",
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"tactic": "ext (_ | _ | _)"
},
{
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"state_before": "case mpr.refine'_3.w.h.some.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫\n (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι).app\n (some WalkingPair.left) =\n ((CommSq.cocone w).ι ≫ (Functor.const WalkingSpan).map αZ).app (some WalkingPair.left)\n\ncase mpr.refine'_3.w.h.some.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY) ≫\n (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι).app\n (some WalkingPair.right) =\n ((CommSq.cocone w).ι ≫ (Functor.const WalkingSpan).map αZ).app (some WalkingPair.right)",
"tactic": "exacts [hh.w.symm, hi.w.symm]"
},
{
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{
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"tactic": "rw [PushoutCocone.condition_zero]"
},
{
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"tactic": "erw [Category.assoc, hh.w, hf.w_assoc]"
},
{
"state_after": "case mpr.refine'_4.id\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni✝ : Y ⟶ Z\nH✝ : IsPushout f g h i✝\nH : IsVanKampenColimit (PushoutCocone.mk h i✝ (_ : f ≫ h = g ≫ i✝))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i✝\nw : CommSq f' g' h' i'\ni : WalkingSpan\n⊢ IsPullback ((span f' g').map (WidePushoutShape.Hom.id i))\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app i)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app i)\n ((span f g).map (WidePushoutShape.Hom.id i))\n\ncase mpr.refine'_4.init.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPullback ((span f' g').map (WidePushoutShape.Hom.init WalkingPair.left))\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app none)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app\n (some WalkingPair.left))\n ((span f g).map (WidePushoutShape.Hom.init WalkingPair.left))\n\ncase mpr.refine'_4.init.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPullback ((span f' g').map (WidePushoutShape.Hom.init WalkingPair.right))\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app none)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app\n (some WalkingPair.right))\n ((span f g).map (WidePushoutShape.Hom.init WalkingPair.right))",
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"tactic": "rintro i _ (_ | _ | _)"
},
{
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"state_before": "case mpr.refine'_4.init.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPullback ((span f' g').map (WidePushoutShape.Hom.init WalkingPair.left))\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app none)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app\n (some WalkingPair.left))\n ((span f g).map (WidePushoutShape.Hom.init WalkingPair.left))\n\ncase mpr.refine'_4.init.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPullback ((span f' g').map (WidePushoutShape.Hom.init WalkingPair.right))\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app none)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app\n (some WalkingPair.right))\n ((span f g).map (WidePushoutShape.Hom.init WalkingPair.right))",
"tactic": "exacts [hf, hg]"
},
{
"state_after": "case mpr.refine'_4.id\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni✝ : Y ⟶ Z\nH✝ : IsPushout f g h i✝\nH : IsVanKampenColimit (PushoutCocone.mk h i✝ (_ : f ≫ h = g ≫ i✝))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i✝\nw : CommSq f' g' h' i'\ni : WalkingSpan\n⊢ IsPullback ((span f' g').map (𝟙 i)) (Option.rec αW (fun val => WalkingPair.rec αX αY val) i)\n (Option.rec αW (fun val => WalkingPair.rec αX αY val) i) ((span f g).map (𝟙 i))",
"state_before": "case mpr.refine'_4.id\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni✝ : Y ⟶ Z\nH✝ : IsPushout f g h i✝\nH : IsVanKampenColimit (PushoutCocone.mk h i✝ (_ : f ≫ h = g ≫ i✝))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i✝\nw : CommSq f' g' h' i'\ni : WalkingSpan\n⊢ IsPullback ((span f' g').map (WidePushoutShape.Hom.id i))\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app i)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app i)\n ((span f g).map (WidePushoutShape.Hom.id i))",
"tactic": "dsimp"
},
{
"state_after": "case mpr.refine'_4.id\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni✝ : Y ⟶ Z\nH✝ : IsPushout f g h i✝\nH : IsVanKampenColimit (PushoutCocone.mk h i✝ (_ : f ≫ h = g ≫ i✝))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i✝\nw : CommSq f' g' h' i'\ni : WalkingSpan\n⊢ IsPullback (𝟙 ((span f' g').obj i)) (Option.rec αW (fun val => WalkingPair.rec αX αY val) i)\n (Option.rec αW (fun val => WalkingPair.rec αX αY val) i) (𝟙 ((span f g).obj i))",
"state_before": "case mpr.refine'_4.id\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni✝ : Y ⟶ Z\nH✝ : IsPushout f g h i✝\nH : IsVanKampenColimit (PushoutCocone.mk h i✝ (_ : f ≫ h = g ≫ i✝))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i✝\nw : CommSq f' g' h' i'\ni : WalkingSpan\n⊢ IsPullback ((span f' g').map (𝟙 i)) (Option.rec αW (fun val => WalkingPair.rec αX αY val) i)\n (Option.rec αW (fun val => WalkingPair.rec αX αY val) i) ((span f g).map (𝟙 i))",
"tactic": "simp_rw [Functor.map_id]"
},
{
"state_after": "no goals",
"state_before": "case mpr.refine'_4.id\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni✝ : Y ⟶ Z\nH✝ : IsPushout f g h i✝\nH : IsVanKampenColimit (PushoutCocone.mk h i✝ (_ : f ≫ h = g ≫ i✝))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i✝\nw : CommSq f' g' h' i'\ni : WalkingSpan\n⊢ IsPullback (𝟙 ((span f' g').obj i)) (Option.rec αW (fun val => WalkingPair.rec αX αY val) i)\n (Option.rec αW (fun val => WalkingPair.rec αX αY val) i) (𝟙 ((span f g).obj i))",
"tactic": "exact IsPullback.of_horiz_isIso ⟨by rw [Category.comp_id, Category.id_comp]⟩"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni✝ : Y ⟶ Z\nH✝ : IsPushout f g h i✝\nH : IsVanKampenColimit (PushoutCocone.mk h i✝ (_ : f ≫ h = g ≫ i✝))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i✝\nw : CommSq f' g' h' i'\ni : WalkingSpan\n⊢ 𝟙 ((span f' g').obj i) ≫ Option.rec αW (fun val => WalkingPair.rec αX αY val) i =\n Option.rec αW (fun val => WalkingPair.rec αX αY val) i ≫ 𝟙 ((span f g).obj i)",
"tactic": "rw [Category.comp_id, Category.id_comp]"
},
{
"state_after": "case mpr.refine'_5.mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (∀ (j : WalkingSpan),\n IsPullback ((CommSq.cocone w).ι.app j)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app j) αZ\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)) →\n IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i\n\ncase mpr.refine'_5.mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i →\n ∀ (j : WalkingSpan),\n IsPullback ((CommSq.cocone w).ι.app j)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app j) αZ\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)",
"state_before": "case mpr.refine'_5\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (∀ (j : WalkingSpan),\n IsPullback ((CommSq.cocone w).ι.app j)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app j) αZ\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)) ↔\n IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i",
"tactic": "constructor"
},
{
"state_after": "case mpr.refine'_5.mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh✝ : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h✝ i\nH : IsVanKampenColimit (PushoutCocone.mk h✝ i (_ : f ≫ h✝ = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h✝\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh :\n ∀ (j : WalkingSpan),\n IsPullback ((CommSq.cocone w).ι.app j)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app j) αZ\n ((PushoutCocone.mk h✝ i (_ : f ≫ h✝ = g ≫ i)).ι.app j)\n⊢ IsPullback h' αX αZ h✝ ∧ IsPullback i' αY αZ i",
"state_before": "case mpr.refine'_5.mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ (∀ (j : WalkingSpan),\n IsPullback ((CommSq.cocone w).ι.app j)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app j) αZ\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)) →\n IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr.refine'_5.mp\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh✝ : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h✝ i\nH : IsVanKampenColimit (PushoutCocone.mk h✝ i (_ : f ≫ h✝ = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h✝\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh :\n ∀ (j : WalkingSpan),\n IsPullback ((CommSq.cocone w).ι.app j)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app j) αZ\n ((PushoutCocone.mk h✝ i (_ : f ≫ h✝ = g ≫ i)).ι.app j)\n⊢ IsPullback h' αX αZ h✝ ∧ IsPullback i' αY αZ i",
"tactic": "exact ⟨h WalkingCospan.left, h WalkingCospan.right⟩"
},
{
"state_after": "case mpr.refine'_5.mpr.intro.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh₁ : IsPullback h' αX αZ h\nh₂ : IsPullback i' αY αZ i\n⊢ IsPullback ((CommSq.cocone w).ι.app none)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app none) αZ\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app none)\n\ncase mpr.refine'_5.mpr.intro.some.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh₁ : IsPullback h' αX αZ h\nh₂ : IsPullback i' αY αZ i\n⊢ IsPullback ((CommSq.cocone w).ι.app (some WalkingPair.left))\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app\n (some WalkingPair.left))\n αZ ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app (some WalkingPair.left))\n\ncase mpr.refine'_5.mpr.intro.some.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh₁ : IsPullback h' αX αZ h\nh₂ : IsPullback i' αY αZ i\n⊢ IsPullback ((CommSq.cocone w).ι.app (some WalkingPair.right))\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app\n (some WalkingPair.right))\n αZ ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app (some WalkingPair.right))",
"state_before": "case mpr.refine'_5.mpr\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i →\n ∀ (j : WalkingSpan),\n IsPullback ((CommSq.cocone w).ι.app j)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app j) αZ\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app j)",
"tactic": "rintro ⟨h₁, h₂⟩ (_ | _ | _)"
},
{
"state_after": "no goals",
"state_before": "case mpr.refine'_5.mpr.intro.some.left\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh₁ : IsPullback h' αX αZ h\nh₂ : IsPullback i' αY αZ i\n⊢ IsPullback ((CommSq.cocone w).ι.app (some WalkingPair.left))\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app\n (some WalkingPair.left))\n αZ ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app (some WalkingPair.left))\n\ncase mpr.refine'_5.mpr.intro.some.right\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh₁ : IsPullback h' αX αZ h\nh₂ : IsPullback i' αY αZ i\n⊢ IsPullback ((CommSq.cocone w).ι.app (some WalkingPair.right))\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app\n (some WalkingPair.right))\n αZ ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app (some WalkingPair.right))",
"tactic": "exacts [h₁, h₂]"
},
{
"state_after": "case mpr.refine'_5.mpr.intro.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh₁ : IsPullback h' αX αZ h\nh₂ : IsPullback i' αY αZ i\n⊢ IsPullback ((CommSq.cocone w).ι.app none) αW αZ (f ≫ h)",
"state_before": "case mpr.refine'_5.mpr.intro.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh₁ : IsPullback h' αX αZ h\nh₂ : IsPullback i' αY αZ i\n⊢ IsPullback ((CommSq.cocone w).ι.app none)\n ((NatTrans.mk fun X_1 => Option.casesOn X_1 αW fun val => WalkingPair.casesOn val αX αY).app none) αZ\n ((PushoutCocone.mk h i (_ : f ≫ h = g ≫ i)).ι.app none)",
"tactic": "dsimp"
},
{
"state_after": "case mpr.refine'_5.mpr.intro.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh₁ : IsPullback h' αX αZ h\nh₂ : IsPullback i' αY αZ i\n⊢ IsPullback (f' ≫ PushoutCocone.inl (CommSq.cocone w)) αW αZ (f ≫ h)",
"state_before": "case mpr.refine'_5.mpr.intro.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh₁ : IsPullback h' αX αZ h\nh₂ : IsPullback i' αY αZ i\n⊢ IsPullback ((CommSq.cocone w).ι.app none) αW αZ (f ≫ h)",
"tactic": "rw [PushoutCocone.condition_zero]"
},
{
"state_after": "no goals",
"state_before": "case mpr.refine'_5.mpr.intro.none\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\nh₁ : IsPullback h' αX αZ h\nh₂ : IsPullback i' αY αZ i\n⊢ IsPullback (f' ≫ PushoutCocone.inl (CommSq.cocone w)) αW αZ (f ≫ h)",
"tactic": "exact hf.paste_horiz h₁"
},
{
"state_after": "no goals",
"state_before": "case mpr.refine'_1\nJ : Type v'\ninst✝¹ : Category J\nC : Type u\ninst✝ : Category C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH✝ : IsPushout f g h i\nH : IsVanKampenColimit (PushoutCocone.mk h i (_ : f ≫ h = g ≫ i))\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX f\nhg : IsPullback g' αW αY g\nhh : CommSq h' αX αZ h\nhi : CommSq i' αY αZ i\nw : CommSq f' g' h' i'\n⊢ IsPushout f' g' h' i' ↔ Nonempty (IsColimit (CommSq.cocone w))",
"tactic": "exact ⟨fun h => h.2, fun h => ⟨w, h⟩⟩"
}
] |
[
118,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/CategoryTheory/Closed/Monoidal.lean
|
CategoryTheory.MonoidalClosed.ofEquiv_uncurry_def
|
[] |
[
344,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.sMod_nonneg
|
[
{
"state_after": "case zero\np : ℕ\nw : 0 < p\n⊢ 0 ≤ 4 % (2 ^ p - 1)\n\ncase succ\np : ℕ\nw : 0 < p\nn✝ : ℕ\n⊢ 0 ≤ (sMod p n✝ ^ 2 - 2) % (2 ^ p - 1)",
"state_before": "p : ℕ\nw : 0 < p\ni : ℕ\n⊢ 0 ≤ sMod p i",
"tactic": "cases i <;> dsimp [sMod]"
},
{
"state_after": "no goals",
"state_before": "case zero\np : ℕ\nw : 0 < p\n⊢ 0 ≤ 4 % (2 ^ p - 1)",
"tactic": "exact sup_eq_right.mp rfl"
},
{
"state_after": "case succ.a\np : ℕ\nw : 0 < p\nn✝ : ℕ\n⊢ 2 ^ p - 1 ≠ 0",
"state_before": "case succ\np : ℕ\nw : 0 < p\nn✝ : ℕ\n⊢ 0 ≤ (sMod p n✝ ^ 2 - 2) % (2 ^ p - 1)",
"tactic": "apply Int.emod_nonneg"
},
{
"state_after": "no goals",
"state_before": "case succ.a\np : ℕ\nw : 0 < p\nn✝ : ℕ\n⊢ 2 ^ p - 1 ≠ 0",
"tactic": "exact mersenne_int_ne_zero p w"
}
] |
[
111,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Algebra/Homology/Additive.lean
|
HomologicalComplex.zsmul_f_apply
|
[] |
[
94,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Involutive.leftInverse
|
[] |
[
890,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
890,
11
] |
Mathlib/Data/Pi/Algebra.lean
|
Pi.mul_comp
|
[] |
[
105,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithBot.coe_bit1
|
[] |
[
573,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
572,
1
] |
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