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/LinearAlgebra/Finrank.lean
|
FiniteDimensional.finrank_le_finrank_of_rank_le_rank
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nV₂ : Type v'\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nh : lift (Module.rank K V) ≤ lift (Module.rank K V₂)\nh' : Module.rank K V₂ < ℵ₀\n⊢ finrank K V ≤ finrank K V₂",
"tactic": "simpa only [toNat_lift] using toNat_le_of_le_of_lt_aleph0 (lift_lt_aleph0.mpr h') h"
}
] |
[
94,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.all_singleton
|
[] |
[
465,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.add_lt_top
|
[] |
[
533,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
9
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
mul_eq_left₀
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.11158\nM₀ : Type u_1\nG₀ : Type ?u.11164\nM₀' : Type ?u.11167\nG₀' : Type ?u.11170\nF : Type ?u.11173\nF' : Type ?u.11176\ninst✝ : CancelMonoidWithZero M₀\na b c : M₀\nha : a ≠ 0\n⊢ a * b = a ↔ b = 1",
"tactic": "rw [Iff.comm, ← mul_right_inj' ha, mul_one]"
}
] |
[
203,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Products.lean
|
TopCat.range_prod_map
|
[
{
"state_after": "case h\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\n⊢ x ∈ Set.range ((forget TopCat).map (prod.map f g)) ↔\n x ∈\n (forget TopCat).map prod.fst ⁻¹' Set.range ((forget TopCat).map f) ∩\n (forget TopCat).map prod.snd ⁻¹' Set.range ((forget TopCat).map g)",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\n⊢ Set.range ((forget TopCat).map (prod.map f g)) =\n (forget TopCat).map prod.fst ⁻¹' Set.range ((forget TopCat).map f) ∩\n (forget TopCat).map prod.snd ⁻¹' Set.range ((forget TopCat).map g)",
"tactic": "ext x"
},
{
"state_after": "case h.mp\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\n⊢ x ∈ Set.range ((forget TopCat).map (prod.map f g)) →\n x ∈\n (forget TopCat).map prod.fst ⁻¹' Set.range ((forget TopCat).map f) ∩\n (forget TopCat).map prod.snd ⁻¹' Set.range ((forget TopCat).map g)\n\ncase h.mpr\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\n⊢ x ∈\n (forget TopCat).map prod.fst ⁻¹' Set.range ((forget TopCat).map f) ∩\n (forget TopCat).map prod.snd ⁻¹' Set.range ((forget TopCat).map g) →\n x ∈ Set.range ((forget TopCat).map (prod.map f g))",
"state_before": "case h\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\n⊢ x ∈ Set.range ((forget TopCat).map (prod.map f g)) ↔\n x ∈\n (forget TopCat).map prod.fst ⁻¹' Set.range ((forget TopCat).map f) ∩\n (forget TopCat).map prod.snd ⁻¹' Set.range ((forget TopCat).map g)",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\ny : (forget TopCat).obj (W ⨯ X)\n⊢ (forget TopCat).map (prod.map f g) y ∈\n (forget TopCat).map prod.fst ⁻¹' Set.range ((forget TopCat).map f) ∩\n (forget TopCat).map prod.snd ⁻¹' Set.range ((forget TopCat).map g)",
"state_before": "case h.mp\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\n⊢ x ∈ Set.range ((forget TopCat).map (prod.map f g)) →\n x ∈\n (forget TopCat).map prod.fst ⁻¹' Set.range ((forget TopCat).map f) ∩\n (forget TopCat).map prod.snd ⁻¹' Set.range ((forget TopCat).map g)",
"tactic": "rintro ⟨y, rfl⟩"
},
{
"state_after": "case h.mp.intro\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\ny : (forget TopCat).obj (W ⨯ X)\n⊢ (∃ y_1, (forget TopCat).map f y_1 = (forget TopCat).map (prod.map f g ≫ prod.fst) y) ∧\n ∃ y_1, (forget TopCat).map g y_1 = (forget TopCat).map (prod.map f g ≫ prod.snd) y",
"state_before": "case h.mp.intro\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\ny : (forget TopCat).obj (W ⨯ X)\n⊢ (forget TopCat).map (prod.map f g) y ∈\n (forget TopCat).map prod.fst ⁻¹' Set.range ((forget TopCat).map f) ∩\n (forget TopCat).map prod.snd ⁻¹' Set.range ((forget TopCat).map g)",
"tactic": "simp only [Set.mem_preimage, Set.mem_range, Set.mem_inter_iff, ← comp_apply]"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\ny : (forget TopCat).obj (W ⨯ X)\n⊢ (∃ y_1, (forget TopCat).map f y_1 = (forget TopCat).map (prod.map f g ≫ prod.fst) y) ∧\n ∃ y_1, (forget TopCat).map g y_1 = (forget TopCat).map (prod.map f g ≫ prod.snd) y",
"tactic": "simp only [Limits.prod.map_fst, Limits.prod.map_snd, exists_apply_eq_apply, comp_apply,\n and_self_iff]"
},
{
"state_after": "case h.mpr.intro.intro.intro\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ x ∈ Set.range ((forget TopCat).map (prod.map f g))",
"state_before": "case h.mpr\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\n⊢ x ∈\n (forget TopCat).map prod.fst ⁻¹' Set.range ((forget TopCat).map f) ∩\n (forget TopCat).map prod.snd ⁻¹' Set.range ((forget TopCat).map g) →\n x ∈ Set.range ((forget TopCat).map (prod.map f g))",
"tactic": "rintro ⟨⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩⟩"
},
{
"state_after": "case h.mpr.intro.intro.intro\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map (prod.map f g) ((forget TopCat).map (prodIsoProd W X).inv (x₁, x₂)) = x",
"state_before": "case h.mpr.intro.intro.intro\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ x ∈ Set.range ((forget TopCat).map (prod.map f g))",
"tactic": "use (prodIsoProd W X).inv (x₁, x₂)"
},
{
"state_after": "case h.mpr.intro.intro.intro.a\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ ∀ (j : Discrete WalkingPair),\n (forget TopCat).map (limit.π (pair Y Z) j)\n ((forget TopCat).map (prod.map f g) ((forget TopCat).map (prodIsoProd W X).inv (x₁, x₂))) =\n (forget TopCat).map (limit.π (pair Y Z) j) x",
"state_before": "case h.mpr.intro.intro.intro\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map (prod.map f g) ((forget TopCat).map (prodIsoProd W X).inv (x₁, x₂)) = x",
"tactic": "apply Concrete.limit_ext"
},
{
"state_after": "case h.mpr.intro.intro.intro.a.mk.left\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left })\n ((forget TopCat).map (prod.map f g) ((forget TopCat).map (prodIsoProd W X).inv (x₁, x₂))) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left }) x\n\ncase h.mpr.intro.intro.intro.a.mk.right\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right })\n ((forget TopCat).map (prod.map f g) ((forget TopCat).map (prodIsoProd W X).inv (x₁, x₂))) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right }) x",
"state_before": "case h.mpr.intro.intro.intro.a\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ ∀ (j : Discrete WalkingPair),\n (forget TopCat).map (limit.π (pair Y Z) j)\n ((forget TopCat).map (prod.map f g) ((forget TopCat).map (prodIsoProd W X).inv (x₁, x₂))) =\n (forget TopCat).map (limit.π (pair Y Z) j) x",
"tactic": "rintro ⟨⟨⟩⟩"
},
{
"state_after": "case h.mpr.intro.intro.intro.a.mk.left\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map ((prodIsoProd W X).inv ≫ prod.map f g ≫ limit.π (pair Y Z) { as := WalkingPair.left }) (x₁, x₂) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left }) x",
"state_before": "case h.mpr.intro.intro.intro.a.mk.left\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left })\n ((forget TopCat).map (prod.map f g) ((forget TopCat).map (prodIsoProd W X).inv (x₁, x₂))) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left }) x",
"tactic": "simp only [← comp_apply, Category.assoc]"
},
{
"state_after": "case h.mpr.intro.intro.intro.a.mk.left\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map ((prodIsoProd W X).inv ≫ prod.fst ≫ f) (x₁, x₂) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left }) x",
"state_before": "case h.mpr.intro.intro.intro.a.mk.left\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map ((prodIsoProd W X).inv ≫ prod.map f g ≫ limit.π (pair Y Z) { as := WalkingPair.left }) (x₁, x₂) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left }) x",
"tactic": "erw [Limits.prod.map_fst]"
},
{
"state_after": "case h.mpr.intro.intro.intro.a.mk.left\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map f ((forget TopCat).map prodFst (x₁, x₂)) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left }) x",
"state_before": "case h.mpr.intro.intro.intro.a.mk.left\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map ((prodIsoProd W X).inv ≫ prod.fst ≫ f) (x₁, x₂) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left }) x",
"tactic": "rw [TopCat.prodIsoProd_inv_fst_assoc,TopCat.comp_app]"
},
{
"state_after": "case h.mpr.intro.intro.intro.a.mk.left\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\nthis : (forget TopCat).map prodFst (x₁, x₂) = x₁\n⊢ (forget TopCat).map f ((forget TopCat).map prodFst (x₁, x₂)) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left }) x",
"state_before": "case h.mpr.intro.intro.intro.a.mk.left\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map f ((forget TopCat).map prodFst (x₁, x₂)) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left }) x",
"tactic": "have : (forget TopCat).map prodFst (x₁, x₂) = x₁ := rfl"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro.intro.a.mk.left\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\nthis : (forget TopCat).map prodFst (x₁, x₂) = x₁\n⊢ (forget TopCat).map f ((forget TopCat).map prodFst (x₁, x₂)) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.left }) x",
"tactic": "rw [this, hx₁]"
},
{
"state_after": "case h.mpr.intro.intro.intro.a.mk.right\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map ((prodIsoProd W X).inv ≫ prod.map f g ≫ limit.π (pair Y Z) { as := WalkingPair.right }) (x₁, x₂) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right }) x",
"state_before": "case h.mpr.intro.intro.intro.a.mk.right\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right })\n ((forget TopCat).map (prod.map f g) ((forget TopCat).map (prodIsoProd W X).inv (x₁, x₂))) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right }) x",
"tactic": "simp only [← comp_apply, Category.assoc]"
},
{
"state_after": "case h.mpr.intro.intro.intro.a.mk.right\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map ((prodIsoProd W X).inv ≫ prod.snd ≫ g) (x₁, x₂) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right }) x",
"state_before": "case h.mpr.intro.intro.intro.a.mk.right\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map ((prodIsoProd W X).inv ≫ prod.map f g ≫ limit.π (pair Y Z) { as := WalkingPair.right }) (x₁, x₂) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right }) x",
"tactic": "erw [Limits.prod.map_snd]"
},
{
"state_after": "case h.mpr.intro.intro.intro.a.mk.right\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map g ((forget TopCat).map prodSnd (x₁, x₂)) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right }) x",
"state_before": "case h.mpr.intro.intro.intro.a.mk.right\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map ((prodIsoProd W X).inv ≫ prod.snd ≫ g) (x₁, x₂) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right }) x",
"tactic": "rw [TopCat.prodIsoProd_inv_snd_assoc,TopCat.comp_app]"
},
{
"state_after": "case h.mpr.intro.intro.intro.a.mk.right\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\nthis : (forget TopCat).map prodSnd (x₁, x₂) = x₂\n⊢ (forget TopCat).map g ((forget TopCat).map prodSnd (x₁, x₂)) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right }) x",
"state_before": "case h.mpr.intro.intro.intro.a.mk.right\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\n⊢ (forget TopCat).map g ((forget TopCat).map prodSnd (x₁, x₂)) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right }) x",
"tactic": "have : (forget TopCat).map prodSnd (x₁, x₂) = x₂ := rfl"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro.intro.a.mk.right\nJ : Type v\ninst✝ : SmallCategory J\nW X Y Z : TopCat\nf : W ⟶ Y\ng : X ⟶ Z\nx : (forget TopCat).obj (Y ⨯ Z)\nx₁ : (forget TopCat).obj W\nhx₁ : (forget TopCat).map f x₁ = (forget TopCat).map prod.fst x\nx₂ : (forget TopCat).obj X\nhx₂ : (forget TopCat).map g x₂ = (forget TopCat).map prod.snd x\nthis : (forget TopCat).map prodSnd (x₁, x₂) = x₂\n⊢ (forget TopCat).map g ((forget TopCat).map prodSnd (x₁, x₂)) =\n (forget TopCat).map (limit.π (pair Y Z) { as := WalkingPair.right }) x",
"tactic": "rw [this, hx₂]"
}
] |
[
274,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.pred_le_self
|
[
{
"state_after": "α : Type ?u.81368\nβ : Type ?u.81371\nγ : Type ?u.81374\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nh : ∃ a, o = succ a\na : Ordinal\ne : o = succ a\n⊢ pred o ≤ o",
"state_before": "α : Type ?u.81368\nβ : Type ?u.81371\nγ : Type ?u.81374\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nh : ∃ a, o = succ a\n⊢ pred o ≤ o",
"tactic": "let ⟨a, e⟩ := h"
},
{
"state_after": "α : Type ?u.81368\nβ : Type ?u.81371\nγ : Type ?u.81374\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nh : ∃ a, o = succ a\na : Ordinal\ne : o = succ a\n⊢ a ≤ succ a",
"state_before": "α : Type ?u.81368\nβ : Type ?u.81371\nγ : Type ?u.81374\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nh : ∃ a, o = succ a\na : Ordinal\ne : o = succ a\n⊢ pred o ≤ o",
"tactic": "rw [e, pred_succ]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.81368\nβ : Type ?u.81371\nγ : Type ?u.81374\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nh : ∃ a, o = succ a\na : Ordinal\ne : o = succ a\n⊢ a ≤ succ a",
"tactic": "exact le_succ a"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.81368\nβ : Type ?u.81371\nγ : Type ?u.81374\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nh : ¬∃ a, o = succ a\n⊢ pred o ≤ o",
"tactic": "rw [pred, dif_neg h]"
}
] |
[
182,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean
|
closure_image_mem_nhds_of_uniformInducing
|
[
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\n⊢ ∃ a, closure (e '' {a' | (a, a') ∈ s}) ∈ 𝓝 b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs : s ∈ 𝓤 α\n⊢ ∃ a, closure (e '' {a' | (a, a') ∈ s}) ∈ 𝓝 b",
"tactic": "obtain ⟨U, ⟨hU, hUo, hsymm⟩, hs⟩ :\n ∃ U, (U ∈ 𝓤 β ∧ IsOpen U ∧ SymmetricRel U) ∧ Prod.map e e ⁻¹' U ⊆ s := by\n rwa [← he₁.comap_uniformity, (uniformity_hasBasis_open_symmetric.comap _).mem_iff] at hs"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\n⊢ ∃ a, closure (e '' {a' | (a, a') ∈ s}) ∈ 𝓝 b",
"state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\n⊢ ∃ a, closure (e '' {a' | (a, a') ∈ s}) ∈ 𝓝 b",
"tactic": "rcases he₂.dense.mem_nhds (UniformSpace.ball_mem_nhds b hU) with ⟨a, ha⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\n⊢ closure (e '' UniformSpace.ball a (Prod.map e e ⁻¹' U)) ∈ 𝓝 b",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\n⊢ ∃ a, closure (e '' {a' | (a, a') ∈ s}) ∈ 𝓝 b",
"tactic": "refine ⟨a, mem_of_superset ?_ (closure_mono <| image_subset _ <| ball_mono hs a)⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\n⊢ closure (e '' UniformSpace.ball a (Prod.map e e ⁻¹' U)) ∈ 𝓝 b",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\n⊢ closure (e '' UniformSpace.ball a (Prod.map e e ⁻¹' U)) ∈ 𝓝 b",
"tactic": "have ho : IsOpen (UniformSpace.ball (e a) U) := UniformSpace.isOpen_ball (e a) hUo"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\ny : β\nhy : y ∈ UniformSpace.ball (e a) U\n⊢ y ∈ closure (e '' UniformSpace.ball a (Prod.map e e ⁻¹' U))",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\n⊢ closure (e '' UniformSpace.ball a (Prod.map e e ⁻¹' U)) ∈ 𝓝 b",
"tactic": "refine mem_of_superset (ho.mem_nhds <| (mem_ball_symmetry hsymm).2 ha) fun y hy => ?_"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\ny : β\nhy : y ∈ UniformSpace.ball (e a) U\nV : Set β\nhV : V ∈ 𝓝 y\n⊢ Set.Nonempty (V ∩ e '' UniformSpace.ball a (Prod.map e e ⁻¹' U))",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\ny : β\nhy : y ∈ UniformSpace.ball (e a) U\n⊢ y ∈ closure (e '' UniformSpace.ball a (Prod.map e e ⁻¹' U))",
"tactic": "refine mem_closure_iff_nhds.2 fun V hV => ?_"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\ny : β\nhy : y ∈ UniformSpace.ball (e a) U\nV : Set β\nhV : V ∈ 𝓝 y\nx : α\nhxV : e x ∈ V\nhxU : e x ∈ UniformSpace.ball (e a) U\n⊢ Set.Nonempty (V ∩ e '' UniformSpace.ball a (Prod.map e e ⁻¹' U))",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\ny : β\nhy : y ∈ UniformSpace.ball (e a) U\nV : Set β\nhV : V ∈ 𝓝 y\n⊢ Set.Nonempty (V ∩ e '' UniformSpace.ball a (Prod.map e e ⁻¹' U))",
"tactic": "rcases he₂.dense.mem_nhds (inter_mem hV (ho.mem_nhds hy)) with ⟨x, hxV, hxU⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs✝ : s ∈ 𝓤 α\nU : Set (β × β)\nhs : Prod.map e e ⁻¹' U ⊆ s\nhU : U ∈ 𝓤 β\nhUo : IsOpen U\nhsymm : SymmetricRel U\na : α\nha : e a ∈ UniformSpace.ball b U\nho : IsOpen (UniformSpace.ball (e a) U)\ny : β\nhy : y ∈ UniformSpace.ball (e a) U\nV : Set β\nhV : V ∈ 𝓝 y\nx : α\nhxV : e x ∈ V\nhxU : e x ∈ UniformSpace.ball (e a) U\n⊢ Set.Nonempty (V ∩ e '' UniformSpace.ball a (Prod.map e e ⁻¹' U))",
"tactic": "exact ⟨e x, hxV, mem_image_of_mem e hxU⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (α × α)\ne : α → β\nb : β\nhe₁ : UniformInducing e\nhe₂ : DenseInducing e\nhs : s ∈ 𝓤 α\n⊢ ∃ U, (U ∈ 𝓤 β ∧ IsOpen U ∧ SymmetricRel U) ∧ Prod.map e e ⁻¹' U ⊆ s",
"tactic": "rwa [← he₁.comap_uniformity, (uniformity_hasBasis_open_symmetric.comap _).mem_iff] at hs"
}
] |
[
258,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.powersetAux_eq_map_coe
|
[] |
[
35,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
LocalizedModule.mk_cancel
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : { x // x ∈ S }\nm : M\n⊢ 1 • 1 • s • m = 1 • s • m",
"tactic": "simp"
}
] |
[
407,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
IsBoundedBilinearMap.continuous_right
|
[] |
[
422,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.strictMono_iff_injective
|
[] |
[
158,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
EsakiaHom.coe_comp
|
[] |
[
328,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/RingTheory/Polynomial/Bernstein.lean
|
bernsteinPolynomial.eval_at_1
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\n⊢ Polynomial.eval 1 (↑(choose n ν) * X ^ ν * (1 - X) ^ (n - ν)) = if ν = n then 1 else 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\n⊢ Polynomial.eval 1 (bernsteinPolynomial R n ν) = if ν = n then 1 else 0",
"tactic": "rw [bernsteinPolynomial]"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ν = n\n⊢ Polynomial.eval 1 (↑(choose n ν) * X ^ ν * (1 - X) ^ (n - ν)) = 1\n\ncase inr\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ¬ν = n\n⊢ Polynomial.eval 1 (↑(choose n ν) * X ^ ν * (1 - X) ^ (n - ν)) = 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\n⊢ Polynomial.eval 1 (↑(choose n ν) * X ^ ν * (1 - X) ^ (n - ν)) = if ν = n then 1 else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝ : CommRing R\nν : ℕ\n⊢ Polynomial.eval 1 (↑(choose ν ν) * X ^ ν * (1 - X) ^ (ν - ν)) = 1",
"state_before": "case inl\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ν = n\n⊢ Polynomial.eval 1 (↑(choose n ν) * X ^ ν * (1 - X) ^ (n - ν)) = 1",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝ : CommRing R\nν : ℕ\n⊢ Polynomial.eval 1 (↑(choose ν ν) * X ^ ν * (1 - X) ^ (ν - ν)) = 1",
"tactic": "simp"
},
{
"state_after": "case inr.inl\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ¬ν = n\nw : n - ν = 0\n⊢ Polynomial.eval 1 (↑(choose n ν) * X ^ ν * (1 - X) ^ (n - ν)) = 0\n\ncase inr.inr\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ¬ν = n\nw : n - ν > 0\n⊢ Polynomial.eval 1 (↑(choose n ν) * X ^ ν * (1 - X) ^ (n - ν)) = 0",
"state_before": "case inr\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ¬ν = n\n⊢ Polynomial.eval 1 (↑(choose n ν) * X ^ ν * (1 - X) ^ (n - ν)) = 0",
"tactic": "obtain w | w := (n - ν).eq_zero_or_pos"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ¬ν = n\nw : n - ν = 0\n⊢ Polynomial.eval 1 (↑(choose n ν) * X ^ ν * (1 - X) ^ (n - ν)) = 0",
"tactic": "simp [Nat.choose_eq_zero_of_lt ((tsub_eq_zero_iff_le.mp w).lt_of_ne (Ne.symm h))]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ¬ν = n\nw : n - ν > 0\n⊢ Polynomial.eval 1 (↑(choose n ν) * X ^ ν * (1 - X) ^ (n - ν)) = 0",
"tactic": "simp [zero_pow w]"
}
] |
[
105,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
AddSubmonoid.smul_mem_pointwise_smul_iff
|
[] |
[
415,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Data/List/Basic.lean
|
List.indexOf_lt_length
|
[] |
[
1209,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1207,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.coe_toAddEquiv
|
[] |
[
358,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Std/Data/String/Lemmas.lean
|
String.prev_of_valid
|
[
{
"state_after": "cs : List Char\nc : Char\ncs' : List Char\n⊢ (if { byteIdx := utf8Len cs + csize c } = 0 then 0\n else utf8PrevAux (cs ++ c :: cs') 0 { byteIdx := utf8Len cs + csize c }) =\n { byteIdx := utf8Len cs }",
"state_before": "cs : List Char\nc : Char\ncs' : List Char\n⊢ prev { data := cs ++ c :: cs' } { byteIdx := utf8Len cs + csize c } = { byteIdx := utf8Len cs }",
"tactic": "simp [prev]"
},
{
"state_after": "cs : List Char\nc : Char\ncs' : List Char\n⊢ utf8PrevAux (cs ++ c :: cs') 0 { byteIdx := utf8Len cs + csize c } = { byteIdx := utf8Len cs }",
"state_before": "cs : List Char\nc : Char\ncs' : List Char\n⊢ (if { byteIdx := utf8Len cs + csize c } = 0 then 0\n else utf8PrevAux (cs ++ c :: cs') 0 { byteIdx := utf8Len cs + csize c }) =\n { byteIdx := utf8Len cs }",
"tactic": "refine (if_neg (Pos.ne_of_gt add_csize_pos)).trans ?_"
},
{
"state_after": "no goals",
"state_before": "cs : List Char\nc : Char\ncs' : List Char\n⊢ utf8PrevAux (cs ++ c :: cs') 0 { byteIdx := utf8Len cs + csize c } = { byteIdx := utf8Len cs }",
"tactic": "rw [utf8PrevAux_of_valid] <;> simp"
}
] |
[
286,
37
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
283,
1
] |
Mathlib/LinearAlgebra/QuotientPi.lean
|
Submodule.quotientPi_aux.right_inv
|
[
{
"state_after": "ι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\n⊢ Function.RightInverse\n ↑(piQuotientLift p (pi Set.univ p) single (_ : ∀ (x : ι), p x ≤ comap (single x) (pi Set.univ p)))\n ↑(quotientPiLift p (fun i => mkQ (p i)) (_ : ∀ (i : ι), p i ≤ ker (mkQ (p i))))",
"state_before": "ι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\n⊢ Function.RightInverse (invFun p) (toFun p)",
"tactic": "dsimp only [toFun, invFun]"
},
{
"state_after": "ι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\n⊢ ↑(comp (quotientPiLift p (fun i => mkQ (p i)) (_ : ∀ (i : ι), p i ≤ ker (mkQ (p i))))\n (piQuotientLift p (pi Set.univ p) single (_ : ∀ (x : ι), p x ≤ comap (single x) (pi Set.univ p)))) =\n ↑LinearMap.id",
"state_before": "ι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\n⊢ Function.RightInverse\n ↑(piQuotientLift p (pi Set.univ p) single (_ : ∀ (x : ι), p x ≤ comap (single x) (pi Set.univ p)))\n ↑(quotientPiLift p (fun i => mkQ (p i)) (_ : ∀ (i : ι), p i ≤ ker (mkQ (p i))))",
"tactic": "rw [Function.rightInverse_iff_comp, ← coe_comp, ← @id_coe R]"
},
{
"state_after": "ι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\n⊢ ↑(comp (quotientPiLift p (fun i => mkQ (p i)) (_ : ∀ (i : ι), p i ≤ ker (mkQ (p i))))\n (piQuotientLift p (pi Set.univ p) single (_ : ∀ (x : ι), p x ≤ comap (single x) (pi Set.univ p))))\n (Pi.single i (Quotient.mk'' x')) j =\n ↑LinearMap.id (Pi.single i (Quotient.mk'' x')) j",
"state_before": "ι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\n⊢ ↑(comp (quotientPiLift p (fun i => mkQ (p i)) (_ : ∀ (i : ι), p i ≤ ker (mkQ (p i))))\n (piQuotientLift p (pi Set.univ p) single (_ : ∀ (x : ι), p x ≤ comap (single x) (pi Set.univ p)))) =\n ↑LinearMap.id",
"tactic": "refine' congr_arg _ (pi_ext fun i x => Quotient.inductionOn' x fun x' => funext fun j => _)"
},
{
"state_after": "ι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\n⊢ (fun i_1 => ↑(mkQ (p i_1)) (↑(single i) x' i_1)) j = Pi.single i (Quotient.mk x') j",
"state_before": "ι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\n⊢ ↑(comp (quotientPiLift p (fun i => mkQ (p i)) (_ : ∀ (i : ι), p i ≤ ker (mkQ (p i))))\n (piQuotientLift p (pi Set.univ p) single (_ : ∀ (x : ι), p x ≤ comap (single x) (pi Set.univ p))))\n (Pi.single i (Quotient.mk'' x')) j =\n ↑LinearMap.id (Pi.single i (Quotient.mk'' x')) j",
"tactic": "rw [comp_apply, piQuotientLift_single, Quotient.mk''_eq_mk, mapQ_apply,\n quotientPiLift_mk, id_apply]"
},
{
"state_after": "case pos\nι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\nhij : i = j\n⊢ Quotient.mk (Pi.single i x' j) = Pi.single i (Quotient.mk x') j\n\ncase neg\nι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\nhij : ¬i = j\n⊢ Quotient.mk (Pi.single i x' j) = Pi.single i (Quotient.mk x') j",
"state_before": "ι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\n⊢ (fun i_1 => ↑(mkQ (p i_1)) (↑(single i) x' i_1)) j = Pi.single i (Quotient.mk x') j",
"tactic": "by_cases hij : i = j <;> simp only [mkQ_apply, coe_single]"
},
{
"state_after": "case pos\nι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\n⊢ Quotient.mk (Pi.single i x' i) = Pi.single i (Quotient.mk x') i",
"state_before": "case pos\nι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\nhij : i = j\n⊢ Quotient.mk (Pi.single i x' j) = Pi.single i (Quotient.mk x') j",
"tactic": "subst hij"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\n⊢ Quotient.mk (Pi.single i x' i) = Pi.single i (Quotient.mk x') i",
"tactic": "rw [Pi.single_eq_same, Pi.single_eq_same]"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type u_1\nR : Type u_3\ninst✝⁸ : CommRing R\nMs : ι → Type u_2\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type ?u.135340\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.135847\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\nhij : ¬i = j\n⊢ Quotient.mk (Pi.single i x' j) = Pi.single i (Quotient.mk x') j",
"tactic": "rw [Pi.single_eq_of_ne (Ne.symm hij), Pi.single_eq_of_ne (Ne.symm hij), Quotient.mk_zero]"
}
] |
[
114,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.ofNat_mul_negSucc'
|
[] |
[
59,
98
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
59,
15
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.IsAtomic.relabel
|
[] |
[
696,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.coeff_C
|
[] |
[
657,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.val_add_one
|
[
{
"state_after": "case inl\nn✝ m n : ℕ\n⊢ ↑(last n + 1) = if last n = last n then 0 else ↑(last n) + 1\n\ncase inr\nn✝ m n : ℕ\ni : Fin (n + 1)\nh : i < last n\n⊢ ↑(i + 1) = if i = last n then 0 else ↑i + 1",
"state_before": "n✝ m n : ℕ\ni : Fin (n + 1)\n⊢ ↑(i + 1) = if i = last n then 0 else ↑i + 1",
"tactic": "rcases(le_last i).eq_or_lt with (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case inl\nn✝ m n : ℕ\n⊢ ↑(last n + 1) = if last n = last n then 0 else ↑(last n) + 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\nn✝ m n : ℕ\ni : Fin (n + 1)\nh : i < last n\n⊢ ↑(i + 1) = if i = last n then 0 else ↑i + 1",
"tactic": "simpa [h.ne] using val_add_one_of_lt h"
}
] |
[
759,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
755,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.sInf_toSubalgebra
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.678241\nR : Type u_2\nA : Type u_1\nB : Type ?u.678250\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : StarRing A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : StarRing B\ninst✝ : StarModule R B\nS : Set (StarSubalgebra R A)\n⊢ ↑(sInf S).toSubalgebra = ↑(sInf (toSubalgebra '' S))",
"tactic": "simp"
}
] |
[
681,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean
|
CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_hom
|
[
{
"state_after": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\ni : d ⟶ e\n⊢ ((α_ f g (h ≫ i)).inv ≫ (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv) ≫ (α_ f g h).inv ▷ i =\n ((α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i) ≫ (α_ f g h).inv ▷ i",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\ni : d ⟶ e\n⊢ (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv = (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i",
"tactic": "rw [← cancel_epi (α_ f g (h ≫ i)).inv, ← cancel_mono ((α_ f g h).inv ▷ i)]"
},
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\ni : d ⟶ e\n⊢ ((α_ f g (h ≫ i)).inv ≫ (α_ f g (h ≫ i)).hom ≫ f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv) ≫ (α_ f g h).inv ▷ i =\n ((α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i) ≫ (α_ f g h).inv ▷ i",
"tactic": "simp"
}
] |
[
297,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.neighborSet_compl
|
[
{
"state_after": "case h\nι : Sort ?u.146524\n𝕜 : Type ?u.146527\nV : Type u\nW : Type v\nX : Type w\nG✝ : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w✝ : V\ne : Sym2 V\nG : SimpleGraph V\nv w : V\n⊢ w ∈ neighborSet (Gᶜ) v ↔ w ∈ neighborSet G vᶜ \\ {v}",
"state_before": "ι : Sort ?u.146524\n𝕜 : Type ?u.146527\nV : Type u\nW : Type v\nX : Type w\nG✝ : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\nG : SimpleGraph V\nv : V\n⊢ neighborSet (Gᶜ) v = neighborSet G vᶜ \\ {v}",
"tactic": "ext w"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Sort ?u.146524\n𝕜 : Type ?u.146527\nV : Type u\nW : Type v\nX : Type w\nG✝ : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w✝ : V\ne : Sym2 V\nG : SimpleGraph V\nv w : V\n⊢ w ∈ neighborSet (Gᶜ) v ↔ w ∈ neighborSet G vᶜ \\ {v}",
"tactic": "simp [and_comm, eq_comm]"
}
] |
[
1003,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1000,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Uniform.lean
|
Pmf.uniformOfFinset_apply_of_not_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5033\nγ : Type ?u.5036\ns : Finset α\nhs : Finset.Nonempty s\na : α\nha : ¬a ∈ s\n⊢ ↑(uniformOfFinset s hs) a = 0",
"tactic": "simp [ha]"
}
] |
[
63,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
Ideal.Quotient.mkₐ_surjective
|
[] |
[
230,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Combinatorics/Quiver/Cast.lean
|
Quiver.Path.cast_cast
|
[
{
"state_after": "U : Type u_1\ninst✝ : Quiver U\nu'' v'' : U\np : Path u'' v''\n⊢ cast (_ : u'' = u'') (_ : v'' = v'') (cast (_ : u'' = u'') (_ : v'' = v'') p) = cast (_ : u'' = u'') (_ : v'' = v'') p",
"state_before": "U : Type u_1\ninst✝ : Quiver U\nu v u' v' u'' v'' : U\np : Path u v\nhu : u = u'\nhv : v = v'\nhu' : u' = u''\nhv' : v' = v''\n⊢ cast hu' hv' (cast hu hv p) = cast (_ : u = u'') (_ : v = v'') p",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "U : Type u_1\ninst✝ : Quiver U\nu'' v'' : U\np : Path u'' v''\n⊢ cast (_ : u'' = u'') (_ : v'' = v'') (cast (_ : u'' = u'') (_ : v'' = v'') p) = cast (_ : u'' = u'') (_ : v'' = v'') p",
"tactic": "rfl"
}
] |
[
106,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/FieldTheory/Subfield.lean
|
Subfield.coe_one
|
[] |
[
402,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/Topology/Constructions.lean
|
continuous_uncurry_left
|
[] |
[
494,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
492,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Cardinal.IsStrongLimit.ne_zero
|
[] |
[
864,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
863,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.Valid.foldl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : α → Char → α\ninit : α\nx✝ : Substring\nh✝ : Valid x✝\nw✝² w✝¹ w✝ : List Char\nh : ValidFor w✝² w✝¹ w✝ x✝\n⊢ Substring.foldl f init x✝ = List.foldl f init (toString x✝).data",
"tactic": "simp [h.foldl, h.toString]"
}
] |
[
1054,
74
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1053,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Ring.lean
|
summable_sum_mul_antidiagonal_of_summable_mul
|
[
{
"state_after": "ι : Type ?u.27646\nκ : Type ?u.27649\nR : Type ?u.27652\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : NonUnitalNonAssocSemiring α\nf g : ℕ → α\ninst✝¹ : T3Space α\ninst✝ : TopologicalSemiring α\nh : Summable fun x => f (↑x.snd).fst * g (↑x.snd).snd\n⊢ Summable fun n => ∑ kl in Nat.antidiagonal n, f kl.fst * g kl.snd",
"state_before": "ι : Type ?u.27646\nκ : Type ?u.27649\nR : Type ?u.27652\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : NonUnitalNonAssocSemiring α\nf g : ℕ → α\ninst✝¹ : T3Space α\ninst✝ : TopologicalSemiring α\nh : Summable fun x => f x.fst * g x.snd\n⊢ Summable fun n => ∑ kl in Nat.antidiagonal n, f kl.fst * g kl.snd",
"tactic": "rw [summable_mul_prod_iff_summable_mul_sigma_antidiagonal] at h"
},
{
"state_after": "ι : Type ?u.27646\nκ : Type ?u.27649\nR : Type ?u.27652\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : NonUnitalNonAssocSemiring α\nf g : ℕ → α\ninst✝¹ : T3Space α\ninst✝ : TopologicalSemiring α\nh : Summable fun x => f (↑x.snd).fst * g (↑x.snd).snd\n⊢ Summable fun x => ∑' (b : ↑↑(Nat.antidiagonal x)), f (↑b).fst * g (↑b).snd",
"state_before": "ι : Type ?u.27646\nκ : Type ?u.27649\nR : Type ?u.27652\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : NonUnitalNonAssocSemiring α\nf g : ℕ → α\ninst✝¹ : T3Space α\ninst✝ : TopologicalSemiring α\nh : Summable fun x => f (↑x.snd).fst * g (↑x.snd).snd\n⊢ Summable fun n => ∑ kl in Nat.antidiagonal n, f kl.fst * g kl.snd",
"tactic": "conv => congr; ext; rw [← Finset.sum_finset_coe, ← tsum_fintype]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.27646\nκ : Type ?u.27649\nR : Type ?u.27652\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : NonUnitalNonAssocSemiring α\nf g : ℕ → α\ninst✝¹ : T3Space α\ninst✝ : TopologicalSemiring α\nh : Summable fun x => f (↑x.snd).fst * g (↑x.snd).snd\n⊢ Summable fun x => ∑' (b : ↑↑(Nat.antidiagonal x)), f (↑b).fst * g (↑b).snd",
"tactic": "exact h.sigma' fun n => (hasSum_fintype _).summable"
}
] |
[
213,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.Injective.of_comp_iff
|
[] |
[
134,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Algebra/Order/Monoid/WithZero/Defs.lean
|
WithZero.zero_le
|
[] |
[
53,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
LocallyConstant.mulIndicator_of_mem
|
[] |
[
558,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
557,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.add_one_lt_exp_of_nonzero
|
[
{
"state_after": "case inl\nx : ℝ\nhx : x ≠ 0\nh : x < 0\n⊢ x + 1 < exp x\n\ncase inr\nx : ℝ\nhx : x ≠ 0\nh : x > 0\n⊢ x + 1 < exp x",
"state_before": "x : ℝ\nhx : x ≠ 0\n⊢ x + 1 < exp x",
"tactic": "cases' lt_or_gt_of_ne hx with h h"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℝ\nhx : x ≠ 0\nh : x > 0\n⊢ x + 1 < exp x",
"tactic": "exact add_one_lt_exp_of_pos h"
},
{
"state_after": "no goals",
"state_before": "case inl\nx : ℝ\nhx : x ≠ 0\nh : x < 0\n⊢ x + 1 < exp x",
"tactic": "exact add_one_lt_exp_of_neg h"
}
] |
[
1984,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1981,
1
] |
Mathlib/Order/CompleteLattice.lean
|
lt_iInf_iff
|
[] |
[
925,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
924,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
lie_add
|
[] |
[
122,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/RingTheory/Finiteness.lean
|
Module.Finite.of_surjective
|
[
{
"state_after": "R : Type u_1\nA : Type ?u.386467\nB : Type ?u.386470\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nhM : Finite R M\nf : M →ₗ[R] N\nhf : Surjective ↑f\n⊢ Submodule.FG (Submodule.map f ⊤)",
"state_before": "R : Type u_1\nA : Type ?u.386467\nB : Type ?u.386470\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nhM : Finite R M\nf : M →ₗ[R] N\nhf : Surjective ↑f\n⊢ Submodule.FG ⊤",
"tactic": "rw [← LinearMap.range_eq_top.2 hf, ← Submodule.map_top]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type ?u.386467\nB : Type ?u.386470\nM : Type u_2\nN : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nhM : Finite R M\nf : M →ₗ[R] N\nhf : Surjective ↑f\n⊢ Submodule.FG (Submodule.map f ⊤)",
"tactic": "exact hM.1.map f"
}
] |
[
563,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
560,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.factorization_prime_le_iff_dvd
|
[
{
"state_after": "d n : ℕ\nhd : d ≠ 0\nhn : n ≠ 0\n⊢ (∀ (p : ℕ), Prime p → ↑(factorization d) p ≤ ↑(factorization n) p) ↔ factorization d ≤ factorization n",
"state_before": "d n : ℕ\nhd : d ≠ 0\nhn : n ≠ 0\n⊢ (∀ (p : ℕ), Prime p → ↑(factorization d) p ≤ ↑(factorization n) p) ↔ d ∣ n",
"tactic": "rw [← factorization_le_iff_dvd hd hn]"
},
{
"state_after": "d n : ℕ\nhd : d ≠ 0\nhn : n ≠ 0\nh : ∀ (p : ℕ), Prime p → ↑(factorization d) p ≤ ↑(factorization n) p\np : ℕ\nhp : ¬Prime p\n⊢ ↑(factorization d) p ≤ ↑(factorization n) p",
"state_before": "d n : ℕ\nhd : d ≠ 0\nhn : n ≠ 0\n⊢ (∀ (p : ℕ), Prime p → ↑(factorization d) p ≤ ↑(factorization n) p) ↔ factorization d ≤ factorization n",
"tactic": "refine' ⟨fun h p => (em p.Prime).elim (h p) fun hp => _, fun h p _ => h p⟩"
},
{
"state_after": "no goals",
"state_before": "d n : ℕ\nhd : d ≠ 0\nhn : n ≠ 0\nh : ∀ (p : ℕ), Prime p → ↑(factorization d) p ≤ ↑(factorization n) p\np : ℕ\nhp : ¬Prime p\n⊢ ↑(factorization d) p ≤ ↑(factorization n) p",
"tactic": "simp_rw [factorization_eq_zero_of_non_prime _ hp, le_refl]"
}
] |
[
466,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
AnalyticOn.sub
|
[] |
[
601,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
599,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.reindexFinsetRange_repr_self
|
[
{
"state_after": "case h.mk\nι : Type u_3\nι' : Type ?u.425289\nR : Type u_1\nR₂ : Type ?u.425295\nK : Type ?u.425298\nM : Type u_2\nM' : Type ?u.425304\nM'' : Type ?u.425307\nV : Type u\nV' : Type ?u.425312\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\ni : ι\nbi : M\nhbi : bi ∈ Finset.image (↑b) Finset.univ\n⊢ ↑(↑(reindexFinsetRange b).repr (↑b i)) { val := bi, property := hbi } =\n ↑(Finsupp.single { val := ↑b i, property := (_ : ↑b i ∈ Finset.image (↑b) Finset.univ) } 1)\n { val := bi, property := hbi }",
"state_before": "ι : Type u_3\nι' : Type ?u.425289\nR : Type u_1\nR₂ : Type ?u.425295\nK : Type ?u.425298\nM : Type u_2\nM' : Type ?u.425304\nM'' : Type ?u.425307\nV : Type u\nV' : Type ?u.425312\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\ni : ι\n⊢ ↑(reindexFinsetRange b).repr (↑b i) =\n Finsupp.single { val := ↑b i, property := (_ : ↑b i ∈ Finset.image (↑b) Finset.univ) } 1",
"tactic": "ext ⟨bi, hbi⟩"
},
{
"state_after": "case h.mk\nι : Type u_3\nι' : Type ?u.425289\nR : Type u_1\nR₂ : Type ?u.425295\nK : Type ?u.425298\nM : Type u_2\nM' : Type ?u.425304\nM'' : Type ?u.425307\nV : Type u\nV' : Type ?u.425312\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\ni : ι\nbi : M\nhbi : bi ∈ Finset.image (↑b) Finset.univ\n⊢ ↑(Finsupp.single { val := ↑b i, property := (_ : ↑b i ∈ range ↑b) } 1)\n (↑(Equiv.subtypeEquiv (Equiv.refl M) (_ : ∀ (a : M), a ∈ range ↑b ↔ a ∈ Finset.image (↑b) Finset.univ)).symm\n { val := bi, property := hbi }) =\n ↑(Finsupp.single { val := ↑b i, property := (_ : ↑b i ∈ Finset.image (↑b) Finset.univ) } 1)\n { val := bi, property := hbi }",
"state_before": "case h.mk\nι : Type u_3\nι' : Type ?u.425289\nR : Type u_1\nR₂ : Type ?u.425295\nK : Type ?u.425298\nM : Type u_2\nM' : Type ?u.425304\nM'' : Type ?u.425307\nV : Type u\nV' : Type ?u.425312\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\ni : ι\nbi : M\nhbi : bi ∈ Finset.image (↑b) Finset.univ\n⊢ ↑(↑(reindexFinsetRange b).repr (↑b i)) { val := bi, property := hbi } =\n ↑(Finsupp.single { val := ↑b i, property := (_ : ↑b i ∈ Finset.image (↑b) Finset.univ) } 1)\n { val := bi, property := hbi }",
"tactic": "rw [reindexFinsetRange, repr_reindex, Finsupp.mapDomain_equiv_apply, reindexRange_repr_self]"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nι : Type u_3\nι' : Type ?u.425289\nR : Type u_1\nR₂ : Type ?u.425295\nK : Type ?u.425298\nM : Type u_2\nM' : Type ?u.425304\nM'' : Type ?u.425307\nV : Type u\nV' : Type ?u.425312\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq M\ni : ι\nbi : M\nhbi : bi ∈ Finset.image (↑b) Finset.univ\n⊢ ↑(Finsupp.single { val := ↑b i, property := (_ : ↑b i ∈ range ↑b) } 1)\n (↑(Equiv.subtypeEquiv (Equiv.refl M) (_ : ∀ (a : M), a ∈ range ↑b ↔ a ∈ Finset.image (↑b) Finset.univ)).symm\n { val := bi, property := hbi }) =\n ↑(Finsupp.single { val := ↑b i, property := (_ : ↑b i ∈ Finset.image (↑b) Finset.univ) } 1)\n { val := bi, property := hbi }",
"tactic": "simp [Finsupp.single_apply]"
}
] |
[
545,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
OrthonormalBasis.map_apply
|
[] |
[
473,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
11
] |
Mathlib/Topology/Algebra/Ring/Basic.lean
|
TopologicalSemiring.continuousNeg_of_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : NonAssocRing α\ninst✝ : ContinuousMul α\n⊢ Continuous fun a => -a",
"tactic": "simpa using (continuous_const.mul continuous_id : Continuous fun x : α => -1 * x)"
}
] |
[
68,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.div_subset_div
|
[] |
[
684,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
683,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval_monomial
|
[] |
[
1138,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1137,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_snd
|
[] |
[
830,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
828,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.integralSum_biUnion_partition
|
[
{
"state_after": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J✝ : Box ι\nπ✝ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : TaggedPrepartition I\nπi : (J : Box ι) → Prepartition J\nhπi : ∀ (J : Box ι), J ∈ π → Prepartition.IsPartition (πi J)\nJ : Box ι\nhJ : J ∈ π.boxes\n⊢ ∑ J' in (πi J).boxes, ↑(↑vol J') (f (tag (biUnionPrepartition π πi) J')) = ↑(↑vol J) (f (tag π J))",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J : Box ι\nπ✝ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : TaggedPrepartition I\nπi : (J : Box ι) → Prepartition J\nhπi : ∀ (J : Box ι), J ∈ π → Prepartition.IsPartition (πi J)\n⊢ integralSum f vol (biUnionPrepartition π πi) = integralSum f vol π",
"tactic": "refine' (π.sum_biUnion_boxes _ _).trans (sum_congr rfl fun J hJ => _)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J✝ : Box ι\nπ✝ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : TaggedPrepartition I\nπi : (J : Box ι) → Prepartition J\nhπi : ∀ (J : Box ι), J ∈ π → Prepartition.IsPartition (πi J)\nJ : Box ι\nhJ : J ∈ π.boxes\n⊢ ∑ J' in (πi J).boxes, ↑(↑vol J') (f (tag (biUnionPrepartition π πi) J')) = ↑(↑vol J) (f (tag π J))",
"tactic": "calc\n (∑ J' in (πi J).boxes, vol J' (f (π.tag <| π.toPrepartition.biUnionIndex πi J'))) =\n ∑ J' in (πi J).boxes, vol J' (f (π.tag J)) :=\n sum_congr rfl fun J' hJ' => by rw [Prepartition.biUnionIndex_of_mem _ hJ hJ']\n _ = vol J (f (π.tag J)) :=\n (vol.map ⟨⟨fun g : E →L[ℝ] F => g (f (π.tag J)), rfl⟩, fun _ _ => rfl⟩).sum_partition_boxes\n le_top (hπi J hJ)"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J✝ : Box ι\nπ✝ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : TaggedPrepartition I\nπi : (J : Box ι) → Prepartition J\nhπi : ∀ (J : Box ι), J ∈ π → Prepartition.IsPartition (πi J)\nJ : Box ι\nhJ : J ∈ π.boxes\nJ' : Box ι\nhJ' : J' ∈ (πi J).boxes\n⊢ ↑(↑vol J') (f (tag π (Prepartition.biUnionIndex π.toPrepartition πi J'))) = ↑(↑vol J') (f (tag π J))",
"tactic": "rw [Prepartition.biUnionIndex_of_mem _ hJ hJ']"
}
] |
[
102,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiff_zero_fun
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.21033\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nx✝ : ↑m ≤ n\n⊢ Differentiable 𝕜 0",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.21033\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nx✝ : ↑m ≤ n\n⊢ Differentiable 𝕜 (iteratedFDeriv 𝕜 m fun x => 0)",
"tactic": "rw [iteratedFDeriv_zero_fun]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.21033\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nm : ℕ\nx✝ : ↑m ≤ n\n⊢ Differentiable 𝕜 0",
"tactic": "exact differentiable_const (0 : E[×m]→L[𝕜] F)"
}
] |
[
84,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Order/BoundedOrder.lean
|
OrderBot.ext_bot
|
[
{
"state_after": "case refl\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25129\nδ : Type ?u.25132\nα : Type u_1\nhA : PartialOrder α\nA B : OrderBot α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ ⊥ = ⊥",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type ?u.25129\nδ : Type ?u.25132\nα : Type u_1\nhA : PartialOrder α\nA : OrderBot α\nhB : PartialOrder α\nB : OrderBot α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ ⊥ = ⊥",
"tactic": "cases PartialOrder.ext H"
},
{
"state_after": "case refl.h\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25129\nδ : Type ?u.25132\nα : Type u_1\nhA : PartialOrder α\nA B : OrderBot α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ ⊥ ≤ ⊥",
"state_before": "case refl\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25129\nδ : Type ?u.25132\nα : Type u_1\nhA : PartialOrder α\nA B : OrderBot α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ ⊥ = ⊥",
"tactic": "apply bot_unique"
},
{
"state_after": "no goals",
"state_before": "case refl.h\nα✝ : Type u\nβ : Type v\nγ : Type ?u.25129\nδ : Type ?u.25132\nα : Type u_1\nhA : PartialOrder α\nA B : OrderBot α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ ⊥ ≤ ⊥",
"tactic": "exact @bot_le _ _ A _"
}
] |
[
439,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Std/Data/Int/DivMod.lean
|
Int.emod_add_ediv
|
[
{
"state_after": "m n : Nat\n⊢ ↑m % ↑(succ n) + -↑(succ n) * -↑(m / succ n) = ↑m",
"state_before": "m n : Nat\n⊢ ofNat m % -[n+1] + -[n+1] * (ofNat m / -[n+1]) = ofNat m",
"tactic": "show (m % succ n + -↑(succ n) * -↑(m / succ n) : Int) = m"
},
{
"state_after": "m n : Nat\n⊢ ↑m % ↑(succ n) + ↑(succ n) * ↑(m / succ n) = ↑m",
"state_before": "m n : Nat\n⊢ ↑m % ↑(succ n) + -↑(succ n) * -↑(m / succ n) = ↑m",
"tactic": "rw [Int.neg_mul_neg]"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\n⊢ ↑m % ↑(succ n) + ↑(succ n) * ↑(m / succ n) = ↑m",
"tactic": "exact congrArg ofNat <| Nat.mod_add_div .."
},
{
"state_after": "a✝ : Nat\n⊢ -[a✝+1] + 0 * (-[a✝+1] / 0) = -[a✝+1]",
"state_before": "a✝ : Nat\n⊢ -[a✝+1] % 0 + 0 * (-[a✝+1] / 0) = -[a✝+1]",
"tactic": "rw [emod_zero]"
},
{
"state_after": "no goals",
"state_before": "a✝ : Nat\n⊢ -[a✝+1] + 0 * (-[a✝+1] / 0) = -[a✝+1]",
"tactic": "rfl"
},
{
"state_after": "x✝¹ x✝ : Int\nm n : Nat\n⊢ -(↑(m % n) + ↑n * ↑(m / n) + 1) = -(↑m + 1)",
"state_before": "x✝¹ x✝ : Int\nm n : Nat\n⊢ ↑n - (↑m % ↑n + 1) - (↑n * (↑m / ↑n) + ↑n) = -[m+1]",
"tactic": "rw [← ofNat_emod, ← ofNat_ediv, ← Int.sub_sub, negSucc_eq, Int.sub_sub n,\n ← Int.neg_neg (_-_), Int.neg_sub, Int.sub_sub_self, Int.add_right_comm]"
},
{
"state_after": "no goals",
"state_before": "x✝¹ x✝ : Int\nm n : Nat\n⊢ -(↑(m % n) + ↑n * ↑(m / n) + 1) = -(↑m + 1)",
"tactic": "exact congrArg (fun x => -(ofNat x + 1)) (Nat.mod_add_div ..)"
}
] |
[
311,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
299,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.integral_deriv_eq_sub'
|
[
{
"state_after": "ι : Type ?u.1862127\n𝕜 : Type ?u.1862130\nE : Type u_1\nF : Type ?u.1862136\nA : Type ?u.1862139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g φ : ℝ → ℝ\nf✝ f' : ℝ → E\na b : ℝ\nf : ℝ → E\nhderiv : deriv f = f'\nhdiff : ∀ (x : ℝ), x ∈ [[a, b]] → DifferentiableAt ℝ f x\nhcont : ContinuousOn f' [[a, b]]\n⊢ IntervalIntegrable (deriv f) volume a b",
"state_before": "ι : Type ?u.1862127\n𝕜 : Type ?u.1862130\nE : Type u_1\nF : Type ?u.1862136\nA : Type ?u.1862139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g φ : ℝ → ℝ\nf✝ f' : ℝ → E\na b : ℝ\nf : ℝ → E\nhderiv : deriv f = f'\nhdiff : ∀ (x : ℝ), x ∈ [[a, b]] → DifferentiableAt ℝ f x\nhcont : ContinuousOn f' [[a, b]]\n⊢ (∫ (y : ℝ) in a..b, f' y) = f b - f a",
"tactic": "rw [← hderiv, integral_deriv_eq_sub hdiff]"
},
{
"state_after": "ι : Type ?u.1862127\n𝕜 : Type ?u.1862130\nE : Type u_1\nF : Type ?u.1862136\nA : Type ?u.1862139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g φ : ℝ → ℝ\nf✝ f' : ℝ → E\na b : ℝ\nf : ℝ → E\nhderiv : deriv f = f'\nhdiff : ∀ (x : ℝ), x ∈ [[a, b]] → DifferentiableAt ℝ f x\nhcont : ContinuousOn f' [[a, b]]\n⊢ IntervalIntegrable f' volume a b",
"state_before": "ι : Type ?u.1862127\n𝕜 : Type ?u.1862130\nE : Type u_1\nF : Type ?u.1862136\nA : Type ?u.1862139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g φ : ℝ → ℝ\nf✝ f' : ℝ → E\na b : ℝ\nf : ℝ → E\nhderiv : deriv f = f'\nhdiff : ∀ (x : ℝ), x ∈ [[a, b]] → DifferentiableAt ℝ f x\nhcont : ContinuousOn f' [[a, b]]\n⊢ IntervalIntegrable (deriv f) volume a b",
"tactic": "rw [hderiv]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1862127\n𝕜 : Type ?u.1862130\nE : Type u_1\nF : Type ?u.1862136\nA : Type ?u.1862139\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g φ : ℝ → ℝ\nf✝ f' : ℝ → E\na b : ℝ\nf : ℝ → E\nhderiv : deriv f = f'\nhdiff : ∀ (x : ℝ), x ∈ [[a, b]] → DifferentiableAt ℝ f x\nhcont : ContinuousOn f' [[a, b]]\n⊢ IntervalIntegrable f' volume a b",
"tactic": "exact hcont.intervalIntegrable"
}
] |
[
1245,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1240,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.toFinsupp_X
|
[] |
[
566,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
565,
1
] |
Mathlib/Topology/Algebra/Monoid.lean
|
ContinuousWithinAt.mul
|
[] |
[
229,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.le_one_iff_exists_coeIdeal
|
[
{
"state_after": "case mp\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\n⊢ J ≤ 1 → ∃ I, ↑I = J\n\ncase mpr\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\n⊢ (∃ I, ↑I = J) → J ≤ 1",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\n⊢ J ≤ 1 ↔ ∃ I, ↑I = J",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ ∃ I, ↑I = J",
"state_before": "case mp\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\n⊢ J ≤ 1 → ∃ I, ↑I = J",
"tactic": "intro hJ"
},
{
"state_after": "case mp.refine'_1\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} → b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}\n\ncase mp.refine'_2\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ 0 ∈ { carrier := {x | ↑(algebraMap R P) x ∈ J}, add_mem' := ?mp.refine'_1 }.carrier\n\ncase mp.refine'_3\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ ∀ (c : R) {x : R},\n x ∈\n { toAddSubsemigroup := { carrier := {x | ↑(algebraMap R P) x ∈ J}, add_mem' := ?mp.refine'_1 },\n zero_mem' := ?mp.refine'_2 }.toAddSubsemigroup.carrier →\n c • x ∈\n { toAddSubsemigroup := { carrier := {x | ↑(algebraMap R P) x ∈ J}, add_mem' := ?mp.refine'_1 },\n zero_mem' := ?mp.refine'_2 }.toAddSubsemigroup.carrier\n\ncase mp.refine'_4\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ ↑{\n toAddSubmonoid :=\n { toAddSubsemigroup := { carrier := {x | ↑(algebraMap R P) x ∈ J}, add_mem' := ?mp.refine'_1 },\n zero_mem' := ?mp.refine'_2 },\n smul_mem' := ?mp.refine'_3 } =\n J",
"state_before": "case mp\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ ∃ I, ↑I = J",
"tactic": "refine' ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, _⟩, _⟩, _⟩, _⟩"
},
{
"state_after": "case mp.refine'_1\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\na b : R\nha : a ∈ {x | ↑(algebraMap R P) x ∈ J}\nhb : b ∈ {x | ↑(algebraMap R P) x ∈ J}\n⊢ a + b ∈ {x | ↑(algebraMap R P) x ∈ J}",
"state_before": "case mp.refine'_1\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} → b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}",
"tactic": "intro a b ha hb"
},
{
"state_after": "case mp.refine'_1\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\na b : R\nha : a ∈ {x | ↑(algebraMap R P) x ∈ J}\nhb : b ∈ {x | ↑(algebraMap R P) x ∈ J}\n⊢ ↑(algebraMap R P) a + ↑(algebraMap R P) b ∈ J",
"state_before": "case mp.refine'_1\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\na b : R\nha : a ∈ {x | ↑(algebraMap R P) x ∈ J}\nhb : b ∈ {x | ↑(algebraMap R P) x ∈ J}\n⊢ a + b ∈ {x | ↑(algebraMap R P) x ∈ J}",
"tactic": "rw [mem_setOf, RingHom.map_add]"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_1\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\na b : R\nha : a ∈ {x | ↑(algebraMap R P) x ∈ J}\nhb : b ∈ {x | ↑(algebraMap R P) x ∈ J}\n⊢ ↑(algebraMap R P) a + ↑(algebraMap R P) b ∈ J",
"tactic": "exact J.val.add_mem ha hb"
},
{
"state_after": "case mp.refine'_2\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ 0 ∈ J",
"state_before": "case mp.refine'_2\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier",
"tactic": "rw [mem_setOf, RingHom.map_zero]"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_2\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ 0 ∈ J",
"tactic": "exact J.val.zero_mem"
},
{
"state_after": "case mp.refine'_3\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nc x : R\nhx :\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier\n⊢ c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier",
"state_before": "case mp.refine'_3\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier",
"tactic": "intro c x hx"
},
{
"state_after": "case mp.refine'_3\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nc x : R\nhx :\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier\n⊢ c • ↑(algebraMap R P) x ∈ J",
"state_before": "case mp.refine'_3\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nc x : R\nhx :\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier\n⊢ c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier",
"tactic": "rw [smul_eq_mul, mem_setOf, RingHom.map_mul, ← Algebra.smul_def]"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_3\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nc x : R\nhx :\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier\n⊢ c • ↑(algebraMap R P) x ∈ J",
"tactic": "exact J.val.smul_mem c hx"
},
{
"state_after": "case mp.refine'_4.a\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nx : P\n⊢ x ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x |\n ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) } ↔\n x ∈ J",
"state_before": "case mp.refine'_4\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\n⊢ ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) } =\n J",
"tactic": "ext x"
},
{
"state_after": "case mp.refine'_4.a.mp\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nx : P\n⊢ x ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x |\n ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) } →\n x ∈ J\n\ncase mp.refine'_4.a.mpr\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nx : P\n⊢ x ∈ J →\n x ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) }",
"state_before": "case mp.refine'_4.a\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nx : P\n⊢ x ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x |\n ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) } ↔\n x ∈ J",
"tactic": "constructor"
},
{
"state_after": "case mp.refine'_4.a.mp.intro.intro\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nx : P\ny : R\nhy :\n y ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) }\neq_y : ↑(Algebra.linearMap R P) y = x\n⊢ x ∈ J",
"state_before": "case mp.refine'_4.a.mp\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nx : P\n⊢ x ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x |\n ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) } →\n x ∈ J",
"tactic": "rintro ⟨y, hy, eq_y⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_4.a.mp.intro.intro\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nx : P\ny : R\nhy :\n y ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) }\neq_y : ↑(Algebra.linearMap R P) y = x\n⊢ x ∈ J",
"tactic": "rwa [← eq_y]"
},
{
"state_after": "case mp.refine'_4.a.mpr\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nx : P\nhx : x ∈ J\n⊢ x ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) }",
"state_before": "case mp.refine'_4.a.mpr\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nx : P\n⊢ x ∈ J →\n x ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) }",
"tactic": "intro hx"
},
{
"state_after": "case mp.refine'_4.a.mpr.intro\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\ny : R\nhx : ↑(algebraMap R P) y ∈ J\n⊢ ↑(algebraMap R P) y ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) }",
"state_before": "case mp.refine'_4.a.mpr\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\nx : P\nhx : x ∈ J\n⊢ x ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) }",
"tactic": "obtain ⟨y, rfl⟩ := (mem_one_iff S).mp (hJ hx)"
},
{
"state_after": "no goals",
"state_before": "case mp.refine'_4.a.mpr.intro\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nhJ : J ≤ 1\ny : R\nhx : ↑(algebraMap R P) y ∈ J\n⊢ ↑(algebraMap R P) y ∈\n ↑{\n toAddSubmonoid :=\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) }.carrier) },\n smul_mem' :=\n (_ :\n ∀ (c : R) {x : R},\n x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier →\n c • x ∈\n {\n toAddSubsemigroup :=\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} → a + b ∈ {x | ↑(algebraMap R P) x ∈ J}) },\n zero_mem' :=\n (_ :\n 0 ∈\n { carrier := {x | ↑(algebraMap R P) x ∈ J},\n add_mem' :=\n (_ :\n ∀ {a b : R},\n a ∈ {x | ↑(algebraMap R P) x ∈ J} →\n b ∈ {x | ↑(algebraMap R P) x ∈ J} →\n a + b ∈\n {x | ↑(algebraMap R P) x ∈ J}) }.carrier) }.toAddSubsemigroup.carrier) }",
"tactic": "exact mem_setOf.mpr ⟨y, hx, rfl⟩"
},
{
"state_after": "case mpr.intro\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nI : Ideal R\nhI : ↑I = J\n⊢ J ≤ 1",
"state_before": "case mpr\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\n⊢ (∃ I, ↑I = J) → J ≤ 1",
"tactic": "rintro ⟨I, hI⟩"
},
{
"state_after": "case mpr.intro\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nI : Ideal R\nhI : ↑I = J\n⊢ ↑I ≤ 1",
"state_before": "case mpr.intro\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nI : Ideal R\nhI : ↑I = J\n⊢ J ≤ 1",
"tactic": "rw [← hI]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nR : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nJ : FractionalIdeal S P\nI : Ideal R\nhI : ↑I = J\n⊢ ↑I ≤ 1",
"tactic": "apply coeIdeal_le_one"
}
] |
[
680,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/CategoryTheory/Subobject/Basic.lean
|
CategoryTheory.Subobject.lower_iso
|
[] |
[
511,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.permCongr_trans
|
[
{
"state_after": "case H\nα : Sort u\nβ : Sort v\nγ : Sort w\nα' : Type u_1\nβ' : Type u_2\ne : α' ≃ β'\np p' : Perm α'\nx✝ : β'\n⊢ ↑((↑(permCongr e) p).trans (↑(permCongr e) p')) x✝ = ↑(↑(permCongr e) (p.trans p')) x✝",
"state_before": "α : Sort u\nβ : Sort v\nγ : Sort w\nα' : Type u_1\nβ' : Type u_2\ne : α' ≃ β'\np p' : Perm α'\n⊢ (↑(permCongr e) p).trans (↑(permCongr e) p') = ↑(permCongr e) (p.trans p')",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Sort u\nβ : Sort v\nγ : Sort w\nα' : Type u_1\nβ' : Type u_2\ne : α' ≃ β'\np p' : Perm α'\nx✝ : β'\n⊢ ↑((↑(permCongr e) p).trans (↑(permCongr e) p')) x✝ = ↑(↑(permCongr e) (p.trans p')) x✝",
"tactic": "simp only [trans_apply, comp_apply, permCongr_apply, symm_apply_apply]"
}
] |
[
451,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
|
Complex.cpow_def_of_ne_zero
|
[] |
[
44,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.homothety_add
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.819565\nk : Type u_3\nV1 : Type u_1\nP1 : Type u_2\nV2 : Type ?u.819577\ninst✝⁵ : CommRing k\ninst✝⁴ : AddCommGroup V1\ninst✝³ : AffineSpace V1 P1\ninst✝² : AddCommGroup V2\ninst✝¹ : Module k V1\ninst✝ : Module k V2\nc : P1\nr₁ r₂ : k\n⊢ homothety c (r₁ + r₂) = r₁ • (id k P1 -ᵥ const k P1 c) +ᵥ homothety c r₂",
"tactic": "simp only [homothety_def, add_smul, vadd_vadd]"
}
] |
[
834,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
MeasureTheory.LocallyIntegrable.aestronglyMeasurable
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.127720\nE : Type u_2\nR : Type ?u.127726\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : SecondCountableTopology X\nhf : LocallyIntegrable f\n⊢ AEStronglyMeasurable f μ",
"tactic": "simpa only [restrict_univ] using (locallyIntegrableOn_univ.mpr hf).aestronglyMeasurable"
}
] |
[
202,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Analysis/Complex/Schwarz.lean
|
Complex.dist_le_div_mul_dist_of_mapsTo_ball
|
[
{
"state_after": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf : ℂ → E\nz z₀ : ℂ\nhd : DifferentiableOn ℂ f (ball z R₁)\nh_maps : MapsTo f (ball z R₁) (ball (f z) R₂)\nhz : z ∈ ball z R₁\n⊢ dist (f z) (f z) ≤ R₂ / R₁ * dist z z\n\ncase inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf : ℂ → E\nc z z₀ : ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\nhne : z ≠ c\n⊢ dist (f z) (f c) ≤ R₂ / R₁ * dist z c",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf : ℂ → E\nc z z₀ : ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\n⊢ dist (f z) (f c) ≤ R₂ / R₁ * dist z c",
"tactic": "rcases eq_or_ne z c with (rfl | hne)"
},
{
"state_after": "no goals",
"state_before": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf : ℂ → E\nc z z₀ : ℂ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (ball (f c) R₂)\nhz : z ∈ ball c R₁\nhne : z ≠ c\n⊢ dist (f z) (f c) ≤ R₂ / R₁ * dist z c",
"tactic": "simpa only [dslope_of_ne _ hne, slope_def_module, norm_smul, norm_inv, ← div_eq_inv_mul, ←\n dist_eq_norm, div_le_iff (dist_pos.2 hne)] using norm_dslope_le_div_of_mapsTo_ball hd h_maps hz"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR R₁ R₂ : ℝ\nf : ℂ → E\nz z₀ : ℂ\nhd : DifferentiableOn ℂ f (ball z R₁)\nh_maps : MapsTo f (ball z R₁) (ball (f z) R₂)\nhz : z ∈ ball z R₁\n⊢ dist (f z) (f z) ≤ R₂ / R₁ * dist z z",
"tactic": "simp only [dist_self, mul_zero, le_rfl]"
}
] |
[
160,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Analysis/Convex/Topology.lean
|
segment_subset_closure_openSegment
|
[
{
"state_after": "ι : Type ?u.6034\n𝕜 : Type u_2\nE : Type u_1\ninst✝⁸ : LinearOrderedRing 𝕜\ninst✝⁷ : DenselyOrdered 𝕜\ninst✝⁶ : TopologicalSpace 𝕜\ninst✝⁵ : OrderTopology 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalSpace E\ninst✝² : ContinuousAdd E\ninst✝¹ : Module 𝕜 E\ninst✝ : ContinuousSMul 𝕜 E\nx y : E\n⊢ (fun θ => (1 - θ) • x + θ • y) '' closure (Ioo 0 1) ⊆ closure ((fun θ => (1 - θ) • x + θ • y) '' Ioo 0 1)",
"state_before": "ι : Type ?u.6034\n𝕜 : Type u_2\nE : Type u_1\ninst✝⁸ : LinearOrderedRing 𝕜\ninst✝⁷ : DenselyOrdered 𝕜\ninst✝⁶ : TopologicalSpace 𝕜\ninst✝⁵ : OrderTopology 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalSpace E\ninst✝² : ContinuousAdd E\ninst✝¹ : Module 𝕜 E\ninst✝ : ContinuousSMul 𝕜 E\nx y : E\n⊢ [x-[𝕜]y] ⊆ closure (openSegment 𝕜 x y)",
"tactic": "rw [segment_eq_image, openSegment_eq_image, ← closure_Ioo (zero_ne_one' 𝕜)]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.6034\n𝕜 : Type u_2\nE : Type u_1\ninst✝⁸ : LinearOrderedRing 𝕜\ninst✝⁷ : DenselyOrdered 𝕜\ninst✝⁶ : TopologicalSpace 𝕜\ninst✝⁵ : OrderTopology 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalSpace E\ninst✝² : ContinuousAdd E\ninst✝¹ : Module 𝕜 E\ninst✝ : ContinuousSMul 𝕜 E\nx y : E\n⊢ (fun θ => (1 - θ) • x + θ • y) '' closure (Ioo 0 1) ⊆ closure ((fun θ => (1 - θ) • x + θ • y) '' Ioo 0 1)",
"tactic": "exact image_closure_subset_closure_image (by continuity)"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.6034\n𝕜 : Type u_2\nE : Type u_1\ninst✝⁸ : LinearOrderedRing 𝕜\ninst✝⁷ : DenselyOrdered 𝕜\ninst✝⁶ : TopologicalSpace 𝕜\ninst✝⁵ : OrderTopology 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalSpace E\ninst✝² : ContinuousAdd E\ninst✝¹ : Module 𝕜 E\ninst✝ : ContinuousSMul 𝕜 E\nx y : E\n⊢ Continuous fun θ => (1 - θ) • x + θ • y",
"tactic": "continuity"
}
] |
[
89,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
|
tendsto_rpow_mul_exp_neg_mul_atTop_nhds_0
|
[
{
"state_after": "s b : ℝ\nhb : 0 < b\n⊢ (fun x => exp (b * x) / x ^ s)⁻¹ =ᶠ[atTop] fun x => x ^ s * exp (-b * x)",
"state_before": "s b : ℝ\nhb : 0 < b\n⊢ Tendsto (fun x => x ^ s * exp (-b * x)) atTop (𝓝 0)",
"tactic": "refine' (tendsto_exp_mul_div_rpow_atTop s b hb).inv_tendsto_atTop.congr' _"
},
{
"state_after": "no goals",
"state_before": "s b : ℝ\nhb : 0 < b\n⊢ (fun x => exp (b * x) / x ^ s)⁻¹ =ᶠ[atTop] fun x => x ^ s * exp (-b * x)",
"tactic": "filter_upwards with x using by simp [exp_neg, inv_div, div_eq_mul_inv _ (exp _)]"
},
{
"state_after": "no goals",
"state_before": "s b : ℝ\nhb : 0 < b\nx : ℝ\n⊢ (fun x => exp (b * x) / x ^ s)⁻¹ x = x ^ s * exp (-b * x)",
"tactic": "simp [exp_neg, inv_div, div_eq_mul_inv _ (exp _)]"
}
] |
[
114,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.coeff_one
|
[
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR : Type ?u.271811\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\nk : ℕ\n⊢ Nat.rawCast k = Nat.rawCast 1 * k",
"tactic": "simp"
}
] |
[
741,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
741,
1
] |
Mathlib/Logic/Equiv/List.lean
|
Encodable.length_sortedUniv
|
[] |
[
213,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/CategoryTheory/Limits/ConcreteCategory.lean
|
CategoryTheory.Limits.Concrete.multiequalizer_ext
|
[
{
"state_after": "case a\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type w\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesLimit F (forget C)\nI : MulticospanIndex C\ninst✝¹ : HasMultiequalizer I\ninst✝ : PreservesLimit (MulticospanIndex.multicospan I) (forget C)\nx y : (forget C).obj (multiequalizer I)\nh : ∀ (t : I.L), (forget C).map (Multiequalizer.ι I t) x = (forget C).map (Multiequalizer.ι I t) y\n⊢ ∀ (j : WalkingMulticospan I.fstTo I.sndTo),\n (forget C).map (limit.π (MulticospanIndex.multicospan I) j) x =\n (forget C).map (limit.π (MulticospanIndex.multicospan I) j) y",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type w\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesLimit F (forget C)\nI : MulticospanIndex C\ninst✝¹ : HasMultiequalizer I\ninst✝ : PreservesLimit (MulticospanIndex.multicospan I) (forget C)\nx y : (forget C).obj (multiequalizer I)\nh : ∀ (t : I.L), (forget C).map (Multiequalizer.ι I t) x = (forget C).map (Multiequalizer.ι I t) y\n⊢ x = y",
"tactic": "apply Concrete.limit_ext"
},
{
"state_after": "case a.left\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type w\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesLimit F (forget C)\nI : MulticospanIndex C\ninst✝¹ : HasMultiequalizer I\ninst✝ : PreservesLimit (MulticospanIndex.multicospan I) (forget C)\nx y : (forget C).obj (multiequalizer I)\nh : ∀ (t : I.L), (forget C).map (Multiequalizer.ι I t) x = (forget C).map (Multiequalizer.ι I t) y\na : I.L\n⊢ (forget C).map (limit.π (MulticospanIndex.multicospan I) (WalkingMulticospan.left a)) x =\n (forget C).map (limit.π (MulticospanIndex.multicospan I) (WalkingMulticospan.left a)) y\n\ncase a.right\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type w\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesLimit F (forget C)\nI : MulticospanIndex C\ninst✝¹ : HasMultiequalizer I\ninst✝ : PreservesLimit (MulticospanIndex.multicospan I) (forget C)\nx y : (forget C).obj (multiequalizer I)\nh : ∀ (t : I.L), (forget C).map (Multiequalizer.ι I t) x = (forget C).map (Multiequalizer.ι I t) y\nb : I.R\n⊢ (forget C).map (limit.π (MulticospanIndex.multicospan I) (WalkingMulticospan.right b)) x =\n (forget C).map (limit.π (MulticospanIndex.multicospan I) (WalkingMulticospan.right b)) y",
"state_before": "case a\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type w\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesLimit F (forget C)\nI : MulticospanIndex C\ninst✝¹ : HasMultiequalizer I\ninst✝ : PreservesLimit (MulticospanIndex.multicospan I) (forget C)\nx y : (forget C).obj (multiequalizer I)\nh : ∀ (t : I.L), (forget C).map (Multiequalizer.ι I t) x = (forget C).map (Multiequalizer.ι I t) y\n⊢ ∀ (j : WalkingMulticospan I.fstTo I.sndTo),\n (forget C).map (limit.π (MulticospanIndex.multicospan I) j) x =\n (forget C).map (limit.π (MulticospanIndex.multicospan I) j) y",
"tactic": "rintro (a | b)"
},
{
"state_after": "no goals",
"state_before": "case a.left\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type w\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesLimit F (forget C)\nI : MulticospanIndex C\ninst✝¹ : HasMultiequalizer I\ninst✝ : PreservesLimit (MulticospanIndex.multicospan I) (forget C)\nx y : (forget C).obj (multiequalizer I)\nh : ∀ (t : I.L), (forget C).map (Multiequalizer.ι I t) x = (forget C).map (Multiequalizer.ι I t) y\na : I.L\n⊢ (forget C).map (limit.π (MulticospanIndex.multicospan I) (WalkingMulticospan.left a)) x =\n (forget C).map (limit.π (MulticospanIndex.multicospan I) (WalkingMulticospan.left a)) y",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case a.right\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : ConcreteCategory C\nJ : Type w\ninst✝³ : SmallCategory J\nF : J ⥤ C\ninst✝² : PreservesLimit F (forget C)\nI : MulticospanIndex C\ninst✝¹ : HasMultiequalizer I\ninst✝ : PreservesLimit (MulticospanIndex.multicospan I) (forget C)\nx y : (forget C).obj (multiequalizer I)\nh : ∀ (t : I.L), (forget C).map (Multiequalizer.ι I t) x = (forget C).map (Multiequalizer.ι I t) y\nb : I.R\n⊢ (forget C).map (limit.π (MulticospanIndex.multicospan I) (WalkingMulticospan.right b)) x =\n (forget C).map (limit.π (MulticospanIndex.multicospan I) (WalkingMulticospan.right b)) y",
"tactic": "rw [← limit.w I.multicospan (WalkingMulticospan.Hom.fst b), comp_apply, comp_apply, h]"
}
] |
[
98,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.algebraMap_X
|
[] |
[
1424,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1423,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.not_mem_empty
|
[] |
[
426,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/ModelTheory/LanguageMap.lean
|
FirstOrder.Language.coe_con
|
[] |
[
583,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
582,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.length_coe
|
[] |
[
578,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/Topology/Algebra/Field.lean
|
IsPreconnected.eq_of_sq_eq
|
[
{
"state_after": "case inl\nK : Type ?u.37022\ninst✝⁷ : DivisionRing K\ninst✝⁶ : TopologicalSpace K\nα : Type u_2\n𝕜 : Type u_1\nf g : α → 𝕜\nS : Set α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : T1Space 𝕜\ninst✝² : Field 𝕜\ninst✝¹ : HasContinuousInv₀ 𝕜\ninst✝ : ContinuousMul 𝕜\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0\ny : α\nhy : y ∈ S\nhy' : f y = g y\nx : α\nhx : x ∈ S\nh : EqOn f g S\n⊢ f x = g x\n\ncase inr\nK : Type ?u.37022\ninst✝⁷ : DivisionRing K\ninst✝⁶ : TopologicalSpace K\nα : Type u_2\n𝕜 : Type u_1\nf g : α → 𝕜\nS : Set α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : T1Space 𝕜\ninst✝² : Field 𝕜\ninst✝¹ : HasContinuousInv₀ 𝕜\ninst✝ : ContinuousMul 𝕜\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0\ny : α\nhy : y ∈ S\nhy' : f y = g y\nx : α\nhx : x ∈ S\nh : EqOn f (-g) S\n⊢ f x = g x",
"state_before": "K : Type ?u.37022\ninst✝⁷ : DivisionRing K\ninst✝⁶ : TopologicalSpace K\nα : Type u_2\n𝕜 : Type u_1\nf g : α → 𝕜\nS : Set α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : T1Space 𝕜\ninst✝² : Field 𝕜\ninst✝¹ : HasContinuousInv₀ 𝕜\ninst✝ : ContinuousMul 𝕜\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0\ny : α\nhy : y ∈ S\nhy' : f y = g y\nx : α\nhx : x ∈ S\n⊢ f x = g x",
"tactic": "rcases hS.eq_or_eq_neg_of_sq_eq hf hg @hsq @hg_ne with (h | h)"
},
{
"state_after": "no goals",
"state_before": "case inl\nK : Type ?u.37022\ninst✝⁷ : DivisionRing K\ninst✝⁶ : TopologicalSpace K\nα : Type u_2\n𝕜 : Type u_1\nf g : α → 𝕜\nS : Set α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : T1Space 𝕜\ninst✝² : Field 𝕜\ninst✝¹ : HasContinuousInv₀ 𝕜\ninst✝ : ContinuousMul 𝕜\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0\ny : α\nhy : y ∈ S\nhy' : f y = g y\nx : α\nhx : x ∈ S\nh : EqOn f g S\n⊢ f x = g x",
"tactic": "exact h hx"
},
{
"state_after": "no goals",
"state_before": "case inr\nK : Type ?u.37022\ninst✝⁷ : DivisionRing K\ninst✝⁶ : TopologicalSpace K\nα : Type u_2\n𝕜 : Type u_1\nf g : α → 𝕜\nS : Set α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : T1Space 𝕜\ninst✝² : Field 𝕜\ninst✝¹ : HasContinuousInv₀ 𝕜\ninst✝ : ContinuousMul 𝕜\nhS : IsPreconnected S\nhf : ContinuousOn f S\nhg : ContinuousOn g S\nhsq : EqOn (f ^ 2) (g ^ 2) S\nhg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0\ny : α\nhy : y ∈ S\nhy' : f y = g y\nx : α\nhx : x ∈ S\nh : EqOn f (-g) S\n⊢ f x = g x",
"tactic": "rw [h _, Pi.neg_apply, neg_eq_iff_add_eq_zero, ← two_mul, mul_eq_zero,\n iff_false_iff.2 (hg_ne _)] at hy' ⊢ <;> assumption"
}
] |
[
164,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean
|
riesz_content_aux_image_nonempty
|
[
{
"state_after": "X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nK : Compacts X\n⊢ Set.Nonempty {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x}",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nK : Compacts X\n⊢ Set.Nonempty (↑Λ '' {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x})",
"tactic": "rw [nonempty_image_iff]"
},
{
"state_after": "X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nK : Compacts X\n⊢ 1 ∈ {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x}",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nK : Compacts X\n⊢ Set.Nonempty {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x}",
"tactic": "use (1 : X →ᵇ ℝ≥0)"
},
{
"state_after": "X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nK : Compacts X\nx : X\na✝ : x ∈ K\n⊢ 1 ≤ ↑1 x",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nK : Compacts X\n⊢ 1 ∈ {f | ∀ (x : X), x ∈ K → 1 ≤ ↑f x}",
"tactic": "intro x _"
},
{
"state_after": "X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nK : Compacts X\nx : X\na✝ : x ∈ K\n⊢ 1 ≤ 1",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nK : Compacts X\nx : X\na✝ : x ∈ K\n⊢ 1 ≤ ↑1 x",
"tactic": "simp only [BoundedContinuousFunction.coe_one, Pi.one_apply]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\nΛ : (X →ᵇ ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\nK : Compacts X\nx : X\na✝ : x ∈ K\n⊢ 1 ≤ 1",
"tactic": "rfl"
}
] |
[
60,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.bit_to_nat
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.402487\nb : Bool\nn : Num\n⊢ ↑(bit b n) = Nat.bit b ↑n",
"tactic": "cases b <;> cases n <;> rfl"
}
] |
[
718,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
718,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean
|
Equiv.Perm.signAux_swap_zero_one'
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na : (_ : Fin (n + 2)) × Fin (n + 2)\nha₁ : a ∈ finPairsLT (n + 2)\nha₂ : ¬a ∈ {{ fst := 1, snd := 0 }}\n⊢ (if ↑(swap 0 1) a.fst ≤ ↑(swap 0 1) a.snd then -1 else 1) = 1",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\n⊢ signAux (swap 0 1) = ∏ x in {{ fst := 1, snd := 0 }}, if ↑(swap 0 1) x.fst ≤ ↑(swap 0 1) x.snd then -1 else 1",
"tactic": "refine' Eq.symm (prod_subset (fun ⟨x₁, x₂⟩ => by\n simp (config := { contextual := true }) [mem_finPairsLT, Fin.one_pos]) fun a ha₁ ha₂ => _)"
},
{
"state_after": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₁ : { fst := a₁, snd := a₂ } ∈ finPairsLT (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\n⊢ (if ↑(swap 0 1) { fst := a₁, snd := a₂ }.fst ≤ ↑(swap 0 1) { fst := a₁, snd := a₂ }.snd then -1 else 1) = 1",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na : (_ : Fin (n + 2)) × Fin (n + 2)\nha₁ : a ∈ finPairsLT (n + 2)\nha₂ : ¬a ∈ {{ fst := 1, snd := 0 }}\n⊢ (if ↑(swap 0 1) a.fst ≤ ↑(swap 0 1) a.snd then -1 else 1) = 1",
"tactic": "rcases a with ⟨a₁, a₂⟩"
},
{
"state_after": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\n⊢ (if ↑(swap 0 1) { fst := a₁, snd := a₂ }.fst ≤ ↑(swap 0 1) { fst := a₁, snd := a₂ }.snd then -1 else 1) = 1",
"state_before": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₁ : { fst := a₁, snd := a₂ } ∈ finPairsLT (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\n⊢ (if ↑(swap 0 1) { fst := a₁, snd := a₂ }.fst ≤ ↑(swap 0 1) { fst := a₁, snd := a₂ }.snd then -1 else 1) = 1",
"tactic": "replace ha₁ : a₂ < a₁ := mem_finPairsLT.1 ha₁"
},
{
"state_after": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"state_before": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\n⊢ (if ↑(swap 0 1) { fst := a₁, snd := a₂ }.fst ≤ ↑(swap 0 1) { fst := a₁, snd := a₂ }.snd then -1 else 1) = 1",
"tactic": "dsimp only"
},
{
"state_after": "case mk.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₂ : Fin (n + 2)\nha₂ : ¬{ fst := 0, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < 0\n⊢ (if ↑(swap 0 1) 0 ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1\n\ncase mk.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"state_before": "case mk\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"tactic": "rcases a₁.zero_le.eq_or_lt with (rfl | H)"
},
{
"state_after": "case mk.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₂ : ¬{ fst := a₁, snd := 0 } ∈ {{ fst := 1, snd := 0 }}\nha₁ : 0 < a₁\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) 0 then -1 else 1) = 1\n\ncase mk.inr.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"state_before": "case mk.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"tactic": "rcases a₂.zero_le.eq_or_lt with (rfl | H')"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nx✝ : (_ : Fin (n + 2)) × Fin (n + 2)\nx₁ x₂ : Fin (n + 2)\n⊢ { fst := x₁, snd := x₂ } ∈ {{ fst := 1, snd := 0 }} → { fst := x₁, snd := x₂ } ∈ finPairsLT (n + 2)",
"tactic": "simp (config := { contextual := true }) [mem_finPairsLT, Fin.one_pos]"
},
{
"state_after": "no goals",
"state_before": "case mk.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₂ : Fin (n + 2)\nha₂ : ¬{ fst := 0, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < 0\n⊢ (if ↑(swap 0 1) 0 ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"tactic": "exact absurd a₂.zero_le ha₁.not_le"
},
{
"state_after": "case mk.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) 0 then -1 else 1) = 1",
"state_before": "case mk.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₂ : ¬{ fst := a₁, snd := 0 } ∈ {{ fst := 1, snd := 0 }}\nha₁ : 0 < a₁\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) 0 then -1 else 1) = 1",
"tactic": "simp only [and_true_iff, eq_self_iff_true, heq_iff_eq, mem_singleton, Sigma.mk.inj_iff] at ha₂"
},
{
"state_after": "case mk.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\nthis : 1 < a₁\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) 0 then -1 else 1) = 1",
"state_before": "case mk.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) 0 then -1 else 1) = 1",
"tactic": "have : 1 < a₁ := lt_of_le_of_ne (Nat.succ_le_of_lt ha₁)\n (Ne.symm (by intro h; apply ha₂; simp [h]))"
},
{
"state_after": "case mk.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\nthis : 1 < a₁\nh01 : ↑(swap 0 1) 0 = 1\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) 0 then -1 else 1) = 1",
"state_before": "case mk.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\nthis : 1 < a₁\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) 0 then -1 else 1) = 1",
"tactic": "have h01 : Equiv.swap (0 : Fin (n + 2)) 1 0 = 1 := by simp"
},
{
"state_after": "no goals",
"state_before": "case mk.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\nthis : 1 < a₁\nh01 : ↑(swap 0 1) 0 = 1\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) 0 then -1 else 1) = 1",
"tactic": "rw [swap_apply_of_ne_of_ne (ne_of_gt H) ha₂, h01, if_neg this.not_le]"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\nh : a₁ = 1\n⊢ False",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\n⊢ a₁ ≠ 1",
"tactic": "intro h"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\nh : a₁ = 1\n⊢ a₁ = 1",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\nh : a₁ = 1\n⊢ False",
"tactic": "apply ha₂"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\nh : a₁ = 1\n⊢ a₁ = 1",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nha₁ : 0 < a₁\nha₂ : ¬a₁ = 1\nthis : 1 < a₁\n⊢ ↑(swap 0 1) 0 = 1",
"tactic": "simp"
},
{
"state_after": "case mk.inr.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\nle : 1 ≤ a₂\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"state_before": "case mk.inr.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"tactic": "have le : 1 ≤ a₂ := Nat.succ_le_of_lt H'"
},
{
"state_after": "case mk.inr.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\nle : 1 ≤ a₂\nlt : 1 < a₁\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"state_before": "case mk.inr.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\nle : 1 ≤ a₂\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"tactic": "have lt : 1 < a₁ := le.trans_lt ha₁"
},
{
"state_after": "case mk.inr.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\nle : 1 ≤ a₂\nlt : 1 < a₁\nh01 : ↑(swap 0 1) 1 = 0\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"state_before": "case mk.inr.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\nle : 1 ≤ a₂\nlt : 1 < a₁\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"tactic": "have h01 : Equiv.swap (0 : Fin (n + 2)) 1 1 = 0 := by simp only [swap_apply_right]"
},
{
"state_after": "case mk.inr.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nlt : 1 < a₁\nh01 : ↑(swap 0 1) 1 = 0\nha₂ : ¬{ fst := a₁, snd := 1 } ∈ {{ fst := 1, snd := 0 }}\nha₁ : 1 < a₁\nH' : 0 < 1\nle : 1 ≤ 1\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) 1 then -1 else 1) = 1\n\ncase mk.inr.inr.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\nle : 1 ≤ a₂\nlt : 1 < a₁\nh01 : ↑(swap 0 1) 1 = 0\nlt' : 1 < a₂\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"state_before": "case mk.inr.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\nle : 1 ≤ a₂\nlt : 1 < a₁\nh01 : ↑(swap 0 1) 1 = 0\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"tactic": "rcases le.eq_or_lt with (rfl | lt')"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\nle : 1 ≤ a₂\nlt : 1 < a₁\n⊢ ↑(swap 0 1) 1 = 0",
"tactic": "simp only [swap_apply_right]"
},
{
"state_after": "no goals",
"state_before": "case mk.inr.inr.inl\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ : Fin (n + 2)\nH : 0 < a₁\nlt : 1 < a₁\nh01 : ↑(swap 0 1) 1 = 0\nha₂ : ¬{ fst := a₁, snd := 1 } ∈ {{ fst := 1, snd := 0 }}\nha₁ : 1 < a₁\nH' : 0 < 1\nle : 1 ≤ 1\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) 1 then -1 else 1) = 1",
"tactic": "rw [swap_apply_of_ne_of_ne H.ne' lt.ne', h01, if_neg H.not_le]"
},
{
"state_after": "no goals",
"state_before": "case mk.inr.inr.inr\nα : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\na₁ a₂ : Fin (n + 2)\nha₂ : ¬{ fst := a₁, snd := a₂ } ∈ {{ fst := 1, snd := 0 }}\nha₁ : a₂ < a₁\nH : 0 < a₁\nH' : 0 < a₂\nle : 1 ≤ a₂\nlt : 1 < a₁\nh01 : ↑(swap 0 1) 1 = 0\nlt' : 1 < a₂\n⊢ (if ↑(swap 0 1) a₁ ≤ ↑(swap 0 1) a₂ then -1 else 1) = 1",
"tactic": "rw [swap_apply_of_ne_of_ne (ne_of_gt H) (ne_of_gt lt),\n swap_apply_of_ne_of_ne (ne_of_gt H') (ne_of_gt lt'), if_neg ha₁.not_le]"
}
] |
[
449,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
9
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.unique
|
[] |
[
460,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
1
] |
Mathlib/Data/Multiset/Lattice.lean
|
Multiset.le_sup
|
[] |
[
67,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.coe_starAe
|
[] |
[
728,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
727,
1
] |
Mathlib/Data/Set/Basic.lean
|
Disjoint.union_right
|
[] |
[
2957,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2956,
1
] |
Mathlib/Algebra/FreeAlgebra.lean
|
FreeAlgebra.lift_unique
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommSemiring R\nX : Type u_2\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\ng : FreeAlgebra R X →ₐ[R] A\n⊢ ↑g ∘ ι R = f ↔\n ↑{ toFun := FreeAlgebra.liftAux R, invFun := fun F => ↑F ∘ ι R,\n left_inv := (_ : ∀ (f : X → A), (fun F => ↑F ∘ ι R) (FreeAlgebra.liftAux R f) = f),\n right_inv :=\n (_ : ∀ (F : FreeAlgebra R X →ₐ[R] A), FreeAlgebra.liftAux R ((fun F => ↑F ∘ ι R) F) = F) }.symm\n g =\n f",
"state_before": "R : Type u_1\ninst✝² : CommSemiring R\nX : Type u_2\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\ng : FreeAlgebra R X →ₐ[R] A\n⊢ ↑g ∘ ι R = f ↔ g = ↑(lift R) f",
"tactic": "rw [← (lift R).symm_apply_eq, lift]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommSemiring R\nX : Type u_2\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\ng : FreeAlgebra R X →ₐ[R] A\n⊢ ↑g ∘ ι R = f ↔\n ↑{ toFun := FreeAlgebra.liftAux R, invFun := fun F => ↑F ∘ ι R,\n left_inv := (_ : ∀ (f : X → A), (fun F => ↑F ∘ ι R) (FreeAlgebra.liftAux R f) = f),\n right_inv :=\n (_ : ∀ (F : FreeAlgebra R X →ₐ[R] A), FreeAlgebra.liftAux R ((fun F => ↑F ∘ ι R) F) = F) }.symm\n g =\n f",
"tactic": "rfl"
}
] |
[
356,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.FinStronglyMeasurable.exists_set_sigmaFinite
|
[
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\n⊢ ∃ t, MeasurableSet t ∧ (∀ (x : α), x ∈ tᶜ → f x = 0) ∧ SigmaFinite (Measure.restrict μ t)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nhf : FinStronglyMeasurable f μ\n⊢ ∃ t, MeasurableSet t ∧ (∀ (x : α), x ∈ tᶜ → f x = 0) ∧ SigmaFinite (Measure.restrict μ t)",
"tactic": "rcases hf with ⟨fs, hT_lt_top, h_approx⟩"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\n⊢ ∃ t, MeasurableSet t ∧ (∀ (x : α), x ∈ tᶜ → f x = 0) ∧ SigmaFinite (Measure.restrict μ t)",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\n⊢ ∃ t, MeasurableSet t ∧ (∀ (x : α), x ∈ tᶜ → f x = 0) ∧ SigmaFinite (Measure.restrict μ t)",
"tactic": "let T n := support (fs n)"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\n⊢ ∃ t, MeasurableSet t ∧ (∀ (x : α), x ∈ tᶜ → f x = 0) ∧ SigmaFinite (Measure.restrict μ t)",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\n⊢ ∃ t, MeasurableSet t ∧ (∀ (x : α), x ∈ tᶜ → f x = 0) ∧ SigmaFinite (Measure.restrict μ t)",
"tactic": "have hT_meas : ∀ n, MeasurableSet (T n) := fun n => SimpleFunc.measurableSet_support (fs n)"
},
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ ∃ t, MeasurableSet t ∧ (∀ (x : α), x ∈ tᶜ → f x = 0) ∧ SigmaFinite (Measure.restrict μ t)",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\n⊢ ∃ t, MeasurableSet t ∧ (∀ (x : α), x ∈ tᶜ → f x = 0) ∧ SigmaFinite (Measure.restrict μ t)",
"tactic": "let t := ⋃ n, T n"
},
{
"state_after": "case intro.intro.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ ∀ (x : α), x ∈ tᶜ → f x = 0\n\ncase intro.intro.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ SigmaFinite (Measure.restrict μ t)",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ ∃ t, MeasurableSet t ∧ (∀ (x : α), x ∈ tᶜ → f x = 0) ∧ SigmaFinite (Measure.restrict μ t)",
"tactic": "refine' ⟨t, MeasurableSet.iUnion hT_meas, _, _⟩"
},
{
"state_after": "case intro.intro.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nh_fs_zero : ∀ (n : ℕ) (x : α), x ∈ tᶜ → ↑(fs n) x = 0\n⊢ ∀ (x : α), x ∈ tᶜ → f x = 0",
"state_before": "case intro.intro.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ ∀ (x : α), x ∈ tᶜ → f x = 0",
"tactic": "have h_fs_zero : ∀ n, ∀ x ∈ tᶜ, fs n x = 0 := by\n intro n x hxt\n rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt\n simpa using hxt n"
},
{
"state_after": "case intro.intro.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nh_fs_zero : ∀ (n : ℕ) (x : α), x ∈ tᶜ → ↑(fs n) x = 0\nx : α\nhxt : x ∈ tᶜ\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 0)",
"state_before": "case intro.intro.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nh_fs_zero : ∀ (n : ℕ) (x : α), x ∈ tᶜ → ↑(fs n) x = 0\n⊢ ∀ (x : α), x ∈ tᶜ → f x = 0",
"tactic": "refine' fun x hxt => tendsto_nhds_unique (h_approx x) _"
},
{
"state_after": "case intro.intro.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nh_fs_zero : ∀ (n : ℕ) (x : α), x ∈ tᶜ → ↑(fs n) x = 0\nx : α\nhxt : x ∈ tᶜ\n⊢ Tendsto (fun n => 0) atTop (𝓝 0)",
"state_before": "case intro.intro.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nh_fs_zero : ∀ (n : ℕ) (x : α), x ∈ tᶜ → ↑(fs n) x = 0\nx : α\nhxt : x ∈ tᶜ\n⊢ Tendsto (fun n => ↑(fs n) x) atTop (𝓝 0)",
"tactic": "rw [funext fun n => h_fs_zero n x hxt]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nh_fs_zero : ∀ (n : ℕ) (x : α), x ∈ tᶜ → ↑(fs n) x = 0\nx : α\nhxt : x ∈ tᶜ\n⊢ Tendsto (fun n => 0) atTop (𝓝 0)",
"tactic": "exact tendsto_const_nhds"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nn : ℕ\nx : α\nhxt : x ∈ tᶜ\n⊢ ↑(fs n) x = 0",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ ∀ (n : ℕ) (x : α), x ∈ tᶜ → ↑(fs n) x = 0",
"tactic": "intro n x hxt"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nn : ℕ\nx : α\nhxt : ∀ (x_1 : ℕ), ¬x ∈ T x_1\n⊢ ↑(fs n) x = 0",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nn : ℕ\nx : α\nhxt : x ∈ tᶜ\n⊢ ↑(fs n) x = 0",
"tactic": "rw [Set.mem_compl_iff, Set.mem_iUnion, not_exists] at hxt"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nn : ℕ\nx : α\nhxt : ∀ (x_1 : ℕ), ¬x ∈ T x_1\n⊢ ↑(fs n) x = 0",
"tactic": "simpa using hxt n"
},
{
"state_after": "case intro.intro.refine'_2.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nn : ℕ\n⊢ ↑↑(Measure.restrict μ t) ((fun n => tᶜ ∪ T n) n) < ⊤\n\ncase intro.intro.refine'_2.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ (⋃ (i : ℕ), (fun n => tᶜ ∪ T n) i) = univ",
"state_before": "case intro.intro.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ SigmaFinite (Measure.restrict μ t)",
"tactic": "refine' ⟨⟨⟨fun n => tᶜ ∪ T n, fun _ => trivial, fun n => _, _⟩⟩⟩"
},
{
"state_after": "case intro.intro.refine'_2.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nn : ℕ\n⊢ ↑↑μ (T n ∩ ⋃ (b : ℕ), T b) < ⊤",
"state_before": "case intro.intro.refine'_2.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nn : ℕ\n⊢ ↑↑(Measure.restrict μ t) ((fun n => tᶜ ∪ T n) n) < ⊤",
"tactic": "rw [Measure.restrict_apply' (MeasurableSet.iUnion hT_meas), Set.union_inter_distrib_right,\n Set.compl_inter_self t, Set.empty_union]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2.refine'_1\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\nn : ℕ\n⊢ ↑↑μ (T n ∩ ⋃ (b : ℕ), T b) < ⊤",
"tactic": "exact (measure_mono (Set.inter_subset_left _ _)).trans_lt (hT_lt_top n)"
},
{
"state_after": "case intro.intro.refine'_2.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ (tᶜ ∪ ⋃ (i : ℕ), T i) = univ",
"state_before": "case intro.intro.refine'_2.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ (⋃ (i : ℕ), (fun n => tᶜ ∪ T n) i) = univ",
"tactic": "rw [← Set.union_iUnion (tᶜ) T]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2.refine'_2\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.240396\nι : Type ?u.240399\ninst✝³ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), ↑↑μ (support ↑(fs n)) < ⊤\nh_approx : ∀ (x : α), Tendsto (fun n => ↑(fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n => support ↑(fs n)\nhT_meas : ∀ (n : ℕ), MeasurableSet (T n)\nt : Set α := ⋃ (n : ℕ), T n\n⊢ (tᶜ ∪ ⋃ (i : ℕ), T i) = univ",
"tactic": "exact Set.compl_union_self _"
}
] |
[
1048,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1028,
1
] |
Mathlib/Data/Matrix/Notation.lean
|
Matrix.transpose_empty_rows
|
[] |
[
196,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean
|
summable_abs_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nκ : Type ?u.62135\nα : Type u_1\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : UniformSpace α\ninst✝¹ : UniformAddGroup α\ninst✝ : CompleteSpace α\nf : ι → α\ns : Set ι := {x | 0 ≤ f x}\nh1 : ∀ (x : ↑s), abs (f ↑x) = f ↑x\nh2 : ∀ (x : ↑(sᶜ)), abs (f ↑x) = -f ↑x\n⊢ ((Summable fun x => abs (f ↑x)) ∧ Summable fun x => abs (f ↑x)) ↔ (Summable fun x => f ↑x) ∧ Summable fun x => -f ↑x",
"tactic": "simp only [h1, h2]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nκ : Type ?u.62135\nα : Type u_1\ninst✝³ : LinearOrderedAddCommGroup α\ninst✝² : UniformSpace α\ninst✝¹ : UniformAddGroup α\ninst✝ : CompleteSpace α\nf : ι → α\ns : Set ι := {x | 0 ≤ f x}\nh1 : ∀ (x : ↑s), abs (f ↑x) = f ↑x\nh2 : ∀ (x : ↑(sᶜ)), abs (f ↑x) = -f ↑x\n⊢ ((Summable fun x => f ↑x) ∧ Summable fun x => -f ↑x) ↔ Summable f",
"tactic": "simp only [summable_neg_iff, summable_subtype_and_compl]"
}
] |
[
258,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/CategoryTheory/Extensive.lean
|
CategoryTheory.NatTrans.equifibered_of_isIso
|
[] |
[
72,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.Valid.contains
|
[
{
"state_after": "no goals",
"state_before": "c : Char\nx✝ : Substring\nh✝ : Valid x✝\nw✝² w✝¹ w✝ : List Char\nh : ValidFor w✝² w✝¹ w✝ x✝\n⊢ Substring.contains x✝ c = true ↔ c ∈ (toString x✝).data",
"tactic": "simp [h.contains, h.toString]"
}
] |
[
1066,
77
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1065,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
Submodule.eq_bot_iff
|
[] |
[
95,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
11
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.polar_sub_right
|
[
{
"state_after": "no goals",
"state_before": "S : Type ?u.171562\nR : Type u_1\nR₁ : Type ?u.171568\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : CommRing R₁\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx y y' : M\n⊢ polar (↑Q) x (y - y') = polar (↑Q) x y - polar (↑Q) x y'",
"tactic": "rw [sub_eq_add_neg, sub_eq_add_neg, polar_add_right, polar_neg_right]"
}
] |
[
308,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/RingTheory/Localization/AsSubring.lean
|
Localization.subalgebra.mem_range_mapToFractionRing_iff_ofField
|
[
{
"state_after": "A : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝⁵ : Field K\ninst✝⁴ : Algebra A K\ninst✝³ : IsFractionRing A K\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nx : K\n⊢ (∃ a s hs, x = IsLocalization.mk' K a { val := s, property := (_ : s ∈ A⁰) }) ↔\n ∃ a s x_1, x = ↑(algebraMap A K) a * (↑(algebraMap A K) s)⁻¹",
"state_before": "A : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝⁵ : Field K\ninst✝⁴ : Algebra A K\ninst✝³ : IsFractionRing A K\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nx : K\n⊢ x ∈ AlgHom.range (mapToFractionRing K S B hS) ↔ ∃ a s x_1, x = ↑(algebraMap A K) a * (↑(algebraMap A K) s)⁻¹",
"tactic": "rw [mem_range_mapToFractionRing_iff]"
},
{
"state_after": "case h.e'_2.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_3\nA : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝⁵ : Field K\ninst✝⁴ : Algebra A K\ninst✝³ : IsFractionRing A K\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nx : K\nx✝² x✝¹ : A\nx✝ : x✝¹ ∈ S\n⊢ ↑(algebraMap A K) x✝² * (↑(algebraMap A K) x✝¹)⁻¹ =\n IsLocalization.mk' K x✝² { val := x✝¹, property := (_ : x✝¹ ∈ A⁰) }",
"state_before": "A : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝⁵ : Field K\ninst✝⁴ : Algebra A K\ninst✝³ : IsFractionRing A K\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nx : K\n⊢ (∃ a s hs, x = IsLocalization.mk' K a { val := s, property := (_ : s ∈ A⁰) }) ↔\n ∃ a s x_1, x = ↑(algebraMap A K) a * (↑(algebraMap A K) s)⁻¹",
"tactic": "convert Iff.rfl"
},
{
"state_after": "case h.e'_2.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_3.e_a\nA : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝⁵ : Field K\ninst✝⁴ : Algebra A K\ninst✝³ : IsFractionRing A K\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nx : K\nx✝² x✝¹ : A\nx✝ : x✝¹ ∈ S\n⊢ (↑(algebraMap A K) x✝¹)⁻¹ =\n ↑(↑(IsUnit.liftRight\n (MonoidHom.restrict (Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A⁰ K)) A⁰)\n (_ :\n ∀ (y : { x // x ∈ A⁰ }),\n IsUnit (↑(Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A⁰ K)) ↑y)))\n { val := x✝¹, property := (_ : x✝¹ ∈ A⁰) })⁻¹",
"state_before": "case h.e'_2.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_3\nA : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝⁵ : Field K\ninst✝⁴ : Algebra A K\ninst✝³ : IsFractionRing A K\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nx : K\nx✝² x✝¹ : A\nx✝ : x✝¹ ∈ S\n⊢ ↑(algebraMap A K) x✝² * (↑(algebraMap A K) x✝¹)⁻¹ =\n IsLocalization.mk' K x✝² { val := x✝¹, property := (_ : x✝¹ ∈ A⁰) }",
"tactic": "congr"
},
{
"state_after": "case h.e'_2.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_3.e_a\nA : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝⁵ : Field K\ninst✝⁴ : Algebra A K\ninst✝³ : IsFractionRing A K\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nx : K\nx✝² x✝¹ : A\nx✝ : x✝¹ ∈ S\n⊢ (↑(algebraMap A K) x✝¹)⁻¹ =\n (↑(↑(IsUnit.liftRight\n (MonoidHom.restrict (Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A⁰ K)) A⁰)\n (_ :\n ∀ (y : { x // x ∈ A⁰ }),\n IsUnit (↑(Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A⁰ K)) ↑y)))\n { val := x✝¹, property := (_ : x✝¹ ∈ A⁰) }))⁻¹",
"state_before": "case h.e'_2.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_3.e_a\nA : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝⁵ : Field K\ninst✝⁴ : Algebra A K\ninst✝³ : IsFractionRing A K\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nx : K\nx✝² x✝¹ : A\nx✝ : x✝¹ ∈ S\n⊢ (↑(algebraMap A K) x✝¹)⁻¹ =\n ↑(↑(IsUnit.liftRight\n (MonoidHom.restrict (Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A⁰ K)) A⁰)\n (_ :\n ∀ (y : { x // x ∈ A⁰ }),\n IsUnit (↑(Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A⁰ K)) ↑y)))\n { val := x✝¹, property := (_ : x✝¹ ∈ A⁰) })⁻¹",
"tactic": "rw [Units.val_inv_eq_inv_val]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_2.h.h.e'_2.h.h.e'_2.h.h.e'_3.e_a\nA : Type u_2\nK : Type u_3\ninst✝⁶ : CommRing A\nS : Submonoid A\nhS : S ≤ A⁰\ninst✝⁵ : Field K\ninst✝⁴ : Algebra A K\ninst✝³ : IsFractionRing A K\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsLocalization S B\nx : K\nx✝² x✝¹ : A\nx✝ : x✝¹ ∈ S\n⊢ (↑(algebraMap A K) x✝¹)⁻¹ =\n (↑(↑(IsUnit.liftRight\n (MonoidHom.restrict (Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A⁰ K)) A⁰)\n (_ :\n ∀ (y : { x // x ∈ A⁰ }),\n IsUnit (↑(Submonoid.LocalizationMap.toMap (IsLocalization.toLocalizationMap A⁰ K)) ↑y)))\n { val := x✝¹, property := (_ : x✝¹ ∈ A⁰) }))⁻¹",
"tactic": "rfl"
}
] |
[
126,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/LinearAlgebra/InvariantBasisNumber.lean
|
card_eq_of_linearEquiv
|
[] |
[
195,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
uniformity_mulOpposite
|
[] |
[
1528,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1526,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
Real.volume_Ico
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.117608\ninst✝ : Fintype ι\na b : ℝ\n⊢ ↑↑volume (Ico a b) = ofReal (b - a)",
"tactic": "simp [volume_val]"
}
] |
[
83,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_dite
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.394402\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\nf : (x : α) → p x → β\ng : (x : α) → ¬p x → β\nx : { x // x ∈ filter p s }\n⊢ p ↑x",
"tactic": "simpa using (mem_filter.mp x.2).2"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.394402\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\nf : (x : α) → p x → β\ng : (x : α) → ¬p x → β\nx : { x // x ∈ filter (fun x => ¬p x) s }\n⊢ ¬p ↑x",
"tactic": "simpa using (mem_filter.mp x.2).2"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.394402\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\ns : Finset α\np : α → Prop\nhp : DecidablePred p\nf : (x : α) → p x → β\ng : (x : α) → ¬p x → β\n⊢ (∏ x in s, if hx : p x then f x hx else g x hx) =\n (∏ x in attach (filter p s), f ↑x (_ : p ↑x)) * ∏ x in attach (filter (fun x => ¬p x) s), g ↑x (_ : ¬p ↑x)",
"tactic": "simp [prod_apply_dite _ _ fun x => x]"
}
] |
[
988,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
983,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
Filter.Tendsto.atTop_zsmul_neg_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type u_1\nl : Filter α\nf✝ : α → R\nr : R\ninst✝¹ : LinearOrderedAddCommGroup R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x => f x • r) l atBot",
"tactic": "simpa using hf.atTop_zsmul_const (neg_pos.2 hr)"
}
] |
[
240,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/CategoryTheory/Localization/Construction.lean
|
CategoryTheory.Localization.Construction.uniq
|
[
{
"state_after": "C : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\n⊢ G₁ = G₂\n\ncase h'\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\n⊢ Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂",
"state_before": "C : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\n⊢ G₁ = G₂",
"tactic": "suffices h' : Quotient.functor _ ⋙ G₁ = Quotient.functor _ ⋙ G₂"
},
{
"state_after": "case refine'_1\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\n⊢ ∀ (X : MorphismProperty.Localization W), G₁.obj X = G₂.obj X\n\ncase refine'_2\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\n⊢ ∀ (X Y : MorphismProperty.Localization W) (f : X ⟶ Y),\n G₁.map f = eqToHom (_ : ?m.26212.obj X = ?m.26213.obj X) ≫ G₂.map f ≫ eqToHom (_ : G₂.obj Y = G₁.obj Y)",
"state_before": "C : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\n⊢ G₁ = G₂",
"tactic": "refine' Functor.ext _ _"
},
{
"state_after": "case refine'_1.mk.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\nX : C\n⊢ G₁.obj { as := { obj := X } } = G₂.obj { as := { obj := X } }",
"state_before": "case refine'_1\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\n⊢ ∀ (X : MorphismProperty.Localization W), G₁.obj X = G₂.obj X",
"tactic": "rintro ⟨⟨X⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.mk.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\nX : C\n⊢ G₁.obj { as := { obj := X } } = G₂.obj { as := { obj := X } }",
"tactic": "apply Functor.congr_obj h"
},
{
"state_after": "case refine'_2.mk.mk.mk.mk.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\nX Y : C\nf✝ : { as := { obj := X } } ⟶ { as := { obj := Y } }\nf : { as := { obj := X } }.as ⟶ { as := { obj := Y } }.as\n⊢ G₁.map (Quot.mk (Quotient.CompClosure (relations W)) f) =\n eqToHom (_ : G₁.obj { as := { obj := X } } = G₂.obj { as := { obj := X } }) ≫\n G₂.map (Quot.mk (Quotient.CompClosure (relations W)) f) ≫\n eqToHom (_ : G₂.obj { as := { obj := Y } } = G₁.obj { as := { obj := Y } })",
"state_before": "case refine'_2\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\n⊢ ∀ (X Y : MorphismProperty.Localization W) (f : X ⟶ Y),\n G₁.map f = eqToHom (_ : G₁.obj X = G₂.obj X) ≫ G₂.map f ≫ eqToHom (_ : G₂.obj Y = G₁.obj Y)",
"tactic": "rintro ⟨⟨X⟩⟩ ⟨⟨Y⟩⟩ ⟨f⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.mk.mk.mk.mk.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\nX Y : C\nf✝ : { as := { obj := X } } ⟶ { as := { obj := Y } }\nf : { as := { obj := X } }.as ⟶ { as := { obj := Y } }.as\n⊢ G₁.map (Quot.mk (Quotient.CompClosure (relations W)) f) =\n eqToHom (_ : G₁.obj { as := { obj := X } } = G₂.obj { as := { obj := X } }) ≫\n G₂.map (Quot.mk (Quotient.CompClosure (relations W)) f) ≫\n eqToHom (_ : G₂.obj { as := { obj := Y } } = G₁.obj { as := { obj := Y } })",
"tactic": "apply Functor.congr_hom h'"
},
{
"state_after": "case h'.refine'_1\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\n⊢ (Quotient.functor (relations W) ⋙ G₁).toPrefunctor.obj = (Quotient.functor (relations W) ⋙ G₂).toPrefunctor.obj\n\ncase h'.refine'_2\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\n⊢ ∀ (a b : LocQuiver W) (e : a ⟶ b),\n (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath e) =\n eqToHom (_ : (Quotient.functor (relations W) ⋙ G₁).obj a = (Quotient.functor (relations W) ⋙ G₂).obj a) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath e) ≫\n eqToHom (_ : (Quotient.functor (relations W) ⋙ G₂).obj b = (Quotient.functor (relations W) ⋙ G₁).obj b)",
"state_before": "case h'\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\n⊢ Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂",
"tactic": "refine' Paths.ext_functor _ _"
},
{
"state_after": "case h'.refine'_1.h\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX : Paths (LocQuiver W)\n⊢ (Quotient.functor (relations W) ⋙ G₁).obj X = (Quotient.functor (relations W) ⋙ G₂).obj X",
"state_before": "case h'.refine'_1\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\n⊢ (Quotient.functor (relations W) ⋙ G₁).toPrefunctor.obj = (Quotient.functor (relations W) ⋙ G₂).toPrefunctor.obj",
"tactic": "ext X"
},
{
"state_after": "case h'.refine'_1.h.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nobj✝ : C\n⊢ (Quotient.functor (relations W) ⋙ G₁).obj { obj := obj✝ } = (Quotient.functor (relations W) ⋙ G₂).obj { obj := obj✝ }",
"state_before": "case h'.refine'_1.h\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX : Paths (LocQuiver W)\n⊢ (Quotient.functor (relations W) ⋙ G₁).obj X = (Quotient.functor (relations W) ⋙ G₂).obj X",
"tactic": "cases X"
},
{
"state_after": "no goals",
"state_before": "case h'.refine'_1.h.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nobj✝ : C\n⊢ (Quotient.functor (relations W) ⋙ G₁).obj { obj := obj✝ } = (Quotient.functor (relations W) ⋙ G₂).obj { obj := obj✝ }",
"tactic": "apply Functor.congr_obj h"
},
{
"state_after": "case h'.refine'_2.mk.mk.inl\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nf : { obj := X }.obj ⟶ { obj := Y }.obj\n⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inl f)) =\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := X } =\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := X }) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inl f)) ≫\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := Y } =\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := Y })\n\ncase h'.refine'_2.mk.mk.inr.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw : W w\n⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw })) =\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := X } =\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := X }) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw })) ≫\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := Y } =\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := Y })",
"state_before": "case h'.refine'_2\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\n⊢ ∀ (a b : LocQuiver W) (e : a ⟶ b),\n (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath e) =\n eqToHom (_ : (Quotient.functor (relations W) ⋙ G₁).obj a = (Quotient.functor (relations W) ⋙ G₂).obj a) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath e) ≫\n eqToHom (_ : (Quotient.functor (relations W) ⋙ G₂).obj b = (Quotient.functor (relations W) ⋙ G₁).obj b)",
"tactic": "rintro ⟨X⟩ ⟨Y⟩ (f | ⟨w, hw⟩)"
},
{
"state_after": "no goals",
"state_before": "case h'.refine'_2.mk.mk.inl\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nf : { obj := X }.obj ⟶ { obj := Y }.obj\n⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inl f)) =\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := X } =\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := X }) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inl f)) ≫\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := Y } =\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := Y })",
"tactic": "simpa only using Functor.congr_hom h f"
},
{
"state_after": "case h'.refine'_2.mk.mk.inr.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\n⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) =\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := X } =\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := X }) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) ≫\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := Y } =\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := Y })",
"state_before": "case h'.refine'_2.mk.mk.inr.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw : W w\n⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw })) =\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := X } =\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := X }) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw })) ≫\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := Y } =\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := Y })",
"tactic": "have hw : W.Q.map w = (wIso w hw).hom := rfl"
},
{
"state_after": "case h'.refine'_2.mk.mk.inr.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\nhw' :\n (MorphismProperty.Q W ⋙ G₁).map w =\n eqToHom (_ : (MorphismProperty.Q W ⋙ G₁).obj { obj := Y }.obj = (MorphismProperty.Q W ⋙ G₂).obj { obj := Y }.obj) ≫\n (MorphismProperty.Q W ⋙ G₂).map w ≫\n eqToHom\n (_ : (MorphismProperty.Q W ⋙ G₂).obj { obj := X }.obj = (MorphismProperty.Q W ⋙ G₁).obj { obj := X }.obj)\n⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) =\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := X } =\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := X }) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) ≫\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := Y } =\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := Y })",
"state_before": "case h'.refine'_2.mk.mk.inr.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\n⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) =\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := X } =\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := X }) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) ≫\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := Y } =\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := Y })",
"tactic": "have hw' := Functor.congr_hom h w"
},
{
"state_after": "case h'.refine'_2.mk.mk.inr.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\nhw' :\n G₁.map (wIso w hw✝).hom =\n eqToHom (_ : (MorphismProperty.Q W ⋙ G₁).obj { obj := Y }.obj = (MorphismProperty.Q W ⋙ G₂).obj { obj := Y }.obj) ≫\n G₂.map (wIso w hw✝).hom ≫\n eqToHom\n (_ : (MorphismProperty.Q W ⋙ G₂).obj { obj := X }.obj = (MorphismProperty.Q W ⋙ G₁).obj { obj := X }.obj)\n⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) =\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := X } =\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := X }) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) ≫\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := Y } =\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := Y })",
"state_before": "case h'.refine'_2.mk.mk.inr.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\nhw' :\n (MorphismProperty.Q W ⋙ G₁).map w =\n eqToHom (_ : (MorphismProperty.Q W ⋙ G₁).obj { obj := Y }.obj = (MorphismProperty.Q W ⋙ G₂).obj { obj := Y }.obj) ≫\n (MorphismProperty.Q W ⋙ G₂).map w ≫\n eqToHom\n (_ : (MorphismProperty.Q W ⋙ G₂).obj { obj := X }.obj = (MorphismProperty.Q W ⋙ G₁).obj { obj := X }.obj)\n⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) =\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := X } =\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := X }) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) ≫\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := Y } =\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := Y })",
"tactic": "simp only [Functor.comp_map, hw] at hw'"
},
{
"state_after": "case h'.refine'_2.mk.mk.inr.mk.refine'_1\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\nhw' :\n G₁.map (wIso w hw✝).hom =\n eqToHom (_ : (MorphismProperty.Q W ⋙ G₁).obj { obj := Y }.obj = (MorphismProperty.Q W ⋙ G₂).obj { obj := Y }.obj) ≫\n G₂.map (wIso w hw✝).hom ≫\n eqToHom\n (_ : (MorphismProperty.Q W ⋙ G₂).obj { obj := X }.obj = (MorphismProperty.Q W ⋙ G₁).obj { obj := X }.obj)\n⊢ G₁.obj ((MorphismProperty.Q W).obj Y) = G₂.obj ((MorphismProperty.Q W).obj Y)\n\ncase h'.refine'_2.mk.mk.inr.mk.refine'_2\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\nhw' :\n G₁.map (wIso w hw✝).hom =\n eqToHom (_ : (MorphismProperty.Q W ⋙ G₁).obj { obj := Y }.obj = (MorphismProperty.Q W ⋙ G₂).obj { obj := Y }.obj) ≫\n G₂.map (wIso w hw✝).hom ≫\n eqToHom\n (_ : (MorphismProperty.Q W ⋙ G₂).obj { obj := X }.obj = (MorphismProperty.Q W ⋙ G₁).obj { obj := X }.obj)\n⊢ G₁.obj ((MorphismProperty.Q W).obj X) = G₂.obj ((MorphismProperty.Q W).obj X)",
"state_before": "case h'.refine'_2.mk.mk.inr.mk\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\nhw' :\n G₁.map (wIso w hw✝).hom =\n eqToHom (_ : (MorphismProperty.Q W ⋙ G₁).obj { obj := Y }.obj = (MorphismProperty.Q W ⋙ G₂).obj { obj := Y }.obj) ≫\n G₂.map (wIso w hw✝).hom ≫\n eqToHom\n (_ : (MorphismProperty.Q W ⋙ G₂).obj { obj := X }.obj = (MorphismProperty.Q W ⋙ G₁).obj { obj := X }.obj)\n⊢ (Quotient.functor (relations W) ⋙ G₁).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) =\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := X } =\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := X }) ≫\n (Quotient.functor (relations W) ⋙ G₂).map (Quiver.Hom.toPath (Sum.inr { val := w, property := hw✝ })) ≫\n eqToHom\n (_ :\n (Quotient.functor (relations W) ⋙ G₂).obj { obj := Y } =\n (Quotient.functor (relations W) ⋙ G₁).obj { obj := Y })",
"tactic": "refine' Functor.congr_inv_of_congr_hom _ _ _ _ _ hw'"
},
{
"state_after": "no goals",
"state_before": "case h'.refine'_2.mk.mk.inr.mk.refine'_1\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\nhw' :\n G₁.map (wIso w hw✝).hom =\n eqToHom (_ : (MorphismProperty.Q W ⋙ G₁).obj { obj := Y }.obj = (MorphismProperty.Q W ⋙ G₂).obj { obj := Y }.obj) ≫\n G₂.map (wIso w hw✝).hom ≫\n eqToHom\n (_ : (MorphismProperty.Q W ⋙ G₂).obj { obj := X }.obj = (MorphismProperty.Q W ⋙ G₁).obj { obj := X }.obj)\n⊢ G₁.obj ((MorphismProperty.Q W).obj Y) = G₂.obj ((MorphismProperty.Q W).obj Y)\n\ncase h'.refine'_2.mk.mk.inr.mk.refine'_2\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\nhw' :\n G₁.map (wIso w hw✝).hom =\n eqToHom (_ : (MorphismProperty.Q W ⋙ G₁).obj { obj := Y }.obj = (MorphismProperty.Q W ⋙ G₂).obj { obj := Y }.obj) ≫\n G₂.map (wIso w hw✝).hom ≫\n eqToHom\n (_ : (MorphismProperty.Q W ⋙ G₂).obj { obj := X }.obj = (MorphismProperty.Q W ⋙ G₁).obj { obj := X }.obj)\n⊢ G₁.obj ((MorphismProperty.Q W).obj X) = G₂.obj ((MorphismProperty.Q W).obj X)",
"tactic": "all_goals apply Functor.congr_obj h"
},
{
"state_after": "no goals",
"state_before": "case h'.refine'_2.mk.mk.inr.mk.refine'_2\nC : Type u_1\ninst✝¹ : Category C\nW : MorphismProperty C\nD : Type u_4\ninst✝ : Category D\nG : C ⥤ D\nhG : MorphismProperty.IsInvertedBy W G\nG₁ G₂ : MorphismProperty.Localization W ⥤ D\nh : MorphismProperty.Q W ⋙ G₁ = MorphismProperty.Q W ⋙ G₂\nX Y : C\nw : { obj := Y }.obj ⟶ { obj := X }.obj\nhw✝ : W w\nhw : (MorphismProperty.Q W).map w = (wIso w hw✝).hom\nhw' :\n G₁.map (wIso w hw✝).hom =\n eqToHom (_ : (MorphismProperty.Q W ⋙ G₁).obj { obj := Y }.obj = (MorphismProperty.Q W ⋙ G₂).obj { obj := Y }.obj) ≫\n G₂.map (wIso w hw✝).hom ≫\n eqToHom\n (_ : (MorphismProperty.Q W ⋙ G₂).obj { obj := X }.obj = (MorphismProperty.Q W ⋙ G₁).obj { obj := X }.obj)\n⊢ G₁.obj ((MorphismProperty.Q W).obj X) = G₂.obj ((MorphismProperty.Q W).obj X)",
"tactic": "apply Functor.congr_obj h"
}
] |
[
210,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Analysis/Analytic/Linear.lean
|
ContinuousLinearMap.hasFPowerSeriesAt
|
[] |
[
62,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
11
] |
Mathlib/Algebra/Ring/Regular.lean
|
isLeftRegular_of_non_zero_divisor
|
[
{
"state_after": "α : Type u_1\ninst✝ : NonUnitalNonAssocRing α\nk : α\nh : ∀ (x : α), k * x = 0 → x = 0\nx y : α\nh' : k * x = k * y\n⊢ k * (x - y) = 0",
"state_before": "α : Type u_1\ninst✝ : NonUnitalNonAssocRing α\nk : α\nh : ∀ (x : α), k * x = 0 → x = 0\n⊢ IsLeftRegular k",
"tactic": "refine' fun x y (h' : k * x = k * y) => sub_eq_zero.mp (h _ _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : NonUnitalNonAssocRing α\nk : α\nh : ∀ (x : α), k * x = 0 → x = 0\nx y : α\nh' : k * x = k * y\n⊢ k * (x - y) = 0",
"tactic": "rw [mul_sub, sub_eq_zero, h']"
}
] |
[
26,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
23,
1
] |
Mathlib/CategoryTheory/Monoidal/Opposite.lean
|
CategoryTheory.op_tensorObj
|
[] |
[
189,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Order/LocallyFinite.lean
|
WithBot.Icc_coe_coe
|
[] |
[
1160,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1159,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.arcsin_lt_iff_lt_sin
|
[] |
[
177,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.965333\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x✝ r\nu : Set (F × F)\nhu : u ∈ uniformity F\nx : E\nhx : x ∈ EMetric.ball 0 r\n⊢ ∃ t,\n t ∈ 𝓝[EMetric.ball 0 r] x ∧\n ∀ᶠ (n : ℕ) in atTop,\n ∀ (y : E), y ∈ t → ((fun y => f (x✝ + y)) y, (fun n y => FormalMultilinearSeries.partialSum p n y) n y) ∈ u",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.965333\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\n⊢ TendstoLocallyUniformlyOn (fun n y => FormalMultilinearSeries.partialSum p n y) (fun y => f (x + y)) atTop\n (EMetric.ball 0 r)",
"tactic": "intro u hu x hx"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.965333\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr r'✝ : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x✝ r\nu : Set (F × F)\nhu : u ∈ uniformity F\nx : E\nhx : x ∈ EMetric.ball 0 r\nr' : ℝ≥0\nxr' : edist x 0 < ↑r'\nhr' : ↑r' < r\n⊢ ∃ t,\n t ∈ 𝓝[EMetric.ball 0 r] x ∧\n ∀ᶠ (n : ℕ) in atTop,\n ∀ (y : E), y ∈ t → ((fun y => f (x✝ + y)) y, (fun n y => FormalMultilinearSeries.partialSum p n y) n y) ∈ u",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.965333\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x✝ r\nu : Set (F × F)\nhu : u ∈ uniformity F\nx : E\nhx : x ∈ EMetric.ball 0 r\n⊢ ∃ t,\n t ∈ 𝓝[EMetric.ball 0 r] x ∧\n ∀ᶠ (n : ℕ) in atTop,\n ∀ (y : E), y ∈ t → ((fun y => f (x✝ + y)) y, (fun n y => FormalMultilinearSeries.partialSum p n y) n y) ∈ u",
"tactic": "rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.965333\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr r'✝ : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x✝ r\nu : Set (F × F)\nhu : u ∈ uniformity F\nx : E\nhx : x ∈ EMetric.ball 0 r\nr' : ℝ≥0\nxr' : edist x 0 < ↑r'\nhr' : ↑r' < r\nthis : EMetric.ball 0 ↑r' ∈ 𝓝 x\n⊢ ∃ t,\n t ∈ 𝓝[EMetric.ball 0 r] x ∧\n ∀ᶠ (n : ℕ) in atTop,\n ∀ (y : E), y ∈ t → ((fun y => f (x✝ + y)) y, (fun n y => FormalMultilinearSeries.partialSum p n y) n y) ∈ u",
"state_before": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.965333\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr r'✝ : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x✝ r\nu : Set (F × F)\nhu : u ∈ uniformity F\nx : E\nhx : x ∈ EMetric.ball 0 r\nr' : ℝ≥0\nxr' : edist x 0 < ↑r'\nhr' : ↑r' < r\n⊢ ∃ t,\n t ∈ 𝓝[EMetric.ball 0 r] x ∧\n ∀ᶠ (n : ℕ) in atTop,\n ∀ (y : E), y ∈ t → ((fun y => f (x✝ + y)) y, (fun n y => FormalMultilinearSeries.partialSum p n y) n y) ∈ u",
"tactic": "have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.965333\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr r'✝ : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x✝ r\nu : Set (F × F)\nhu : u ∈ uniformity F\nx : E\nhx : x ∈ EMetric.ball 0 r\nr' : ℝ≥0\nxr' : edist x 0 < ↑r'\nhr' : ↑r' < r\nthis : EMetric.ball 0 ↑r' ∈ 𝓝 x\n⊢ ∀ᶠ (n : ℕ) in atTop,\n ∀ (y : E),\n y ∈ EMetric.ball 0 ↑r' → ((fun y => f (x✝ + y)) y, (fun n y => FormalMultilinearSeries.partialSum p n y) n y) ∈ u",
"state_before": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.965333\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr r'✝ : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x✝ r\nu : Set (F × F)\nhu : u ∈ uniformity F\nx : E\nhx : x ∈ EMetric.ball 0 r\nr' : ℝ≥0\nxr' : edist x 0 < ↑r'\nhr' : ↑r' < r\nthis : EMetric.ball 0 ↑r' ∈ 𝓝 x\n⊢ ∃ t,\n t ∈ 𝓝[EMetric.ball 0 r] x ∧\n ∀ᶠ (n : ℕ) in atTop,\n ∀ (y : E), y ∈ t → ((fun y => f (x✝ + y)) y, (fun n y => FormalMultilinearSeries.partialSum p n y) n y) ∈ u",
"tactic": "refine' ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.965333\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx✝ : E\nr r'✝ : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x✝ r\nu : Set (F × F)\nhu : u ∈ uniformity F\nx : E\nhx : x ∈ EMetric.ball 0 r\nr' : ℝ≥0\nxr' : edist x 0 < ↑r'\nhr' : ↑r' < r\nthis : EMetric.ball 0 ↑r' ∈ 𝓝 x\n⊢ ∀ᶠ (n : ℕ) in atTop,\n ∀ (y : E),\n y ∈ EMetric.ball 0 ↑r' → ((fun y => f (x✝ + y)) y, (fun n y => FormalMultilinearSeries.partialSum p n y) n y) ∈ u",
"tactic": "simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu"
}
] |
[
837,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
830,
1
] |
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