mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X : Type u
inst✝⁶ : MetricSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : Nonempty X
Y : Type v
inst✝³ : MetricSpace Y
inst✝² : CompactSpace Y
inst✝¹ : Nonempty Y
γ : Type w
inst✝ : MetricSpace γ
Φ : X → γ
Ψ : Y → γ
ha : Isometry Φ
hb : Isometry Ψ
xX : X
h✝ : xX ∈ univ
s : Set γ := range Φ ∪ range Ψ
Φ' : X → ↑s := fun y ↦ ⟨Φ y, ⋯⟩
Ψ' : Y → ↑s := fun y ↦ ⟨Ψ y, ⋯⟩
IΦ' : Isometry Φ'
IΨ' : Isometry Ψ'
this✝¹ : IsCompact s
this✝ : CompactSpace ↑s
ΦΦ' : Φ = Subtype.val ∘ Φ'
ΨΨ' : Ψ = Subtype.val ∘ Ψ'
this : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ')
⊢ ghDist X Y ≤ hausdorffDist (range Φ) (range Ψ)

Tactic:
this

Goal state after:
X : Type u
inst✝⁶ : MetricSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : Nonempty X
Y : Type v
inst✝³ : MetricSpace Y
inst✝² : CompactSpace Y
inst✝¹ : Nonempty Y
γ : Type w
inst✝ : MetricSpace γ
Φ : X → γ
Ψ : Y → γ
ha : Isometry Φ
hb : Isometry Ψ
xX : X
h✝ : xX ∈ univ
s : Set γ := range Φ ∪ range Ψ
Φ' : X → ↑s := fun y ↦ ⟨Φ y, ⋯⟩
Ψ' : Y → ↑s := fun y ↦ ⟨Ψ y, ⋯⟩
IΦ' : Isometry Φ'
IΨ' : Isometry Ψ'
this✝¹ : IsCompact s
this✝ : CompactSpace ↑s
ΦΦ' : Φ = Subtype.val ∘ Φ'
ΨΨ' : Ψ = Subtype.val ∘ Ψ'
this : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ')
⊢ ghDist X Y ≤ hausdorffDist (range Φ') (range Ψ')
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
⊢ lowerBasis α = (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ {s | ∃ a, (Ici a)ᶜ = s}}

Tactic:
[

Goal state after:
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
⊢ lowerBasis α = (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ {s | ∃ a, (Ici a)ᶜ = s}}
Module: Mathlib.Topology.Order.UpperLowerSetTopology

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Monotone f
⊢ Continuous[inst✝³, inst✝²] f

Tactic:
[

Goal state after:
case mp
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : Topology.IsUpperSet β
f : α → β
hf : Monotone f
⊢ Continuous[inst✝³, inst✝²] f
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
F : Set (Set α)
hF : F.Finite
this : Finite ↑F
f : { a // a ∈ F } → α
hf : ∀ (x : { a // a ∈ F }), (Ici (f x))ᶜ = ↑x
⊢ ⋂ i ∈ range f, (Ici i)ᶜ = ⋂₀ F

Tactic:
[

Goal state after:
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
F : Set (Set α)
hF : F.Finite
this : Finite ↑F
f : { a // a ∈ F } → α
hf : ∀ (x : { a // a ∈ F }), (Ici (f x))ᶜ = ↑x
⊢ ⋂ i ∈ range f, (Ici i)ᶜ = ⋂₀ F
Module: Mathlib.Topology.Order.UpperLowerSetTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : IsUpper β
f : α → β
hf : Monotone f
⊢ Continuous[inst✝³, inst✝²] f

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : Topology.IsUpperSet α
inst✝ : IsUpper β
f : α → β
hf : Monotone f
⊢ Continuous[inst✝³, inst✝²] f
Module: Mathlib.Topology.Order.UpperLowerSetTopology

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a b : α
⊢ toUpperSet a ⤳ toUpperSet b ↔ b ≤ a

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : Preorder α
a b : α
⊢ toUpperSet a ⤳ toUpperSet b ↔ b ≤ a
Module: Mathlib.Topology.Order.UpperLowerSetTopology

Goal state before:
pp: α : Type u_1
inst✝ : Preorder α
a b : α
⊢ toLowerSet a ⤳ toLowerSet b ↔ a ≤ b

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : Preorder α
a b : α
⊢ toLowerSet a ⤳ toLowerSet b ↔ a ≤ b
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
s : Set α
hs : s.Finite
t : Set β
ht : t.Finite
⊢ IsOpen[lower (α × β)] ((↑(upperClosure s))ᶜ ×ˢ (↑(upperClosure t))ᶜ)

Tactic:
[

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝⁷ : Preorder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : IsLower α
inst✝⁴ : OrderBot α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsLower β
inst✝ : OrderBot β
s : Set α
hs : s.Finite
t : Set β
ht : t.Finite
⊢ IsOpen[lower (α × β)] ((↑(upperClosure s))ᶜ ×ˢ (↑(upperClosure t))ᶜ)
Module: Mathlib.Topology.Order.LowerUpperTopology

Goal state before:
pp: case mp
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ IsUpper αᵒᵈ → IsLower α

pp: case mpr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ IsLower α → IsUpper αᵒᵈ

Tactic:
· apply OrderDual.instIsLower

Goal state after:
case mpr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ IsLower α → IsUpper αᵒᵈ
Module: Mathlib.Topology.Order.Category.FrameAdjunction

Goal state before:
pp: case h
L : Type u_1
inst✝ : CompleteLattice L
u v : L
⊢ {x | x (u ⊓ v)} = {x | x u} ∩ {x | x v}

Tactic:
[

Goal state after:
case h
L : Type u_1
inst✝ : CompleteLattice L
u v : L
⊢ {x | x (u ⊓ v)} = {x | x u} ∩ {x | x v}
Module: Mathlib.Topology.Order.Category.FrameAdjunction

Goal state before:
pp: L : Type u_1
inst✝ : CompleteLattice L
⊢ ∀ (s t : Set (PT L)), (∃ u, {x | x u} = s) → (∃ u, {x | x u} = t) → ∃ u, {x | x u} = s ∩ t

Tactic:
rintro s t ⟨u, rfl⟩ ⟨v, rfl⟩; use u ⊓ v; simp_rw [map_inf]; rfl

Goal state after:
Module: Mathlib.Topology.Order.Category.FrameAdjunction

Goal state before:
pp: case h
L : Type u_1
inst✝ : CompleteLattice L
S : Set (Set (PT L))
f : (t : Set (PT L)) → t ∈ S → L
hf : ∀ (t : Set (PT L)) (a : t ∈ S), {x | x (f t a)} = t
⊢ {x | x (⨆ t, ⨆ (ht : t ∈ S), f t ht)} = ⋃₀ S

Tactic:
[

Goal state after:
case h
L : Type u_1
inst✝ : CompleteLattice L
S : Set (Set (PT L))
f : (t : Set (PT L)) → t ∈ S → L
hf : ∀ (t : Set (PT L)) (a : t ∈ S), {x | x (f t a)} = t
⊢ {x | x (⨆ t, ⨆ (ht : t ∈ S), f t ht)} = ⋃₀ S
Module: Mathlib.Topology.Order.HullKernel

Goal state before:
pp: case cons
α : Type u_1
inst✝¹ : SemilatticeInf α
T : Set α
inst✝ : OrderTop α
hT : ∀ p ∈ T, InfPrime p
a : α
F' : Finset α
h✝ : a ∉ F'
I4 : hull T (F'.inf id) = T ↓∩ ⋃ a ∈ ↑F', Set.Ici a
⊢ hull T ((cons a F' h✝).inf id) = T ↓∩ ⋃ a_1 ∈ ↑(cons a F' h✝), Set.Ici a_1

Tactic:
simp [hull_inf hT, I4]

Goal state after:
Module: Mathlib.Topology.Order.HullKernel

Goal state before:
pp: case right
α : Type u_1
inst✝² : CompleteLattice α
T : Set α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
hT : ∀ p ∈ T, InfPrime p
hG : OrderGenerates T
S : Set ↑T
⊢ hull T (kernel S) ⊆ ⋂₀ {t | IsClosed[instTopologicalSpaceSubtype] t ∧ S ⊆ t}

Tactic:
[

Goal state after:
case right
α : Type u_1
inst✝² : CompleteLattice α
T : Set α
inst✝¹ : TopologicalSpace α
inst✝ : IsLower α
hT : ∀ p ∈ T, InfPrime p
hG : OrderGenerates T
S : Set ↑T
⊢ hull T (kernel S) ⊆ ⋂₀ {t | IsClosed[instTopologicalSpaceSubtype] t ∧ S ⊆ t}
Module: Mathlib.Topology.Order.LawsonTopology

Goal state before:
pp: α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLawson α
⊢ {s | ∃ t, t.Finite ∧ ∃ u, IsOpen[scott α univ] u ∧ u \ ↑(upperClosure t) = s} =
    {c |
      ∃ a ∈ {s | ∃ t, t.Finite ∧ (↑(upperClosure t))ᶜ = s},
        ∃ b ∈ {U | IsOpen[scott α univ] U}, ⇑WithLower.toLower ⁻¹' a ∩ ⇑WithScott.toScott ⁻¹' b = c}

Tactic:
[

Goal state after:
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLawson α
⊢ {s | ∃ t, t.Finite ∧ ∃ u, IsOpen[scott α univ] u ∧ u \ ↑(upperClosure t) = s} =
    {c |
      ∃ a ∈ {s | ∃ t, t.Finite ∧ (↑(upperClosure t))ᶜ = s},
        ∃ b ∈ {U | IsOpen[scott α univ] U}, ⇑WithLower.toLower ⁻¹' a ∩ ⇑WithScott.toScott ⁻¹' b = c}
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun n ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun n ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
this : range Φ = range Ψ
⊢ (↑(range Ψ) ≃ᵢ y.Rep) = (↑(range Φ) ≃ᵢ y.Rep)

Tactic:
this

Goal state after:
x y : GHSpace
hxy : dist x y = 0
Φ : x.Rep → ↥(lp (fun n ↦ ℝ) ∞)
Ψ : y.Rep → ↥(lp (fun n ↦ ℝ) ∞)
Φisom : Isometry Φ
Ψisom : Isometry Ψ
DΦΨ : 0 = hausdorffDist (range Φ) (range Ψ)
this : range Φ = range Ψ
⊢ (↑(range Ψ) ≃ᵢ y.Rep) = (↑(range Ψ) ≃ᵢ y.Rep)
Module: Mathlib.Topology.Separation.PerfectlyNormal

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
h : ∀ (s : Set X), IsClosed[inst✝] s → ∃ f, s = ⇑f ⁻¹' {0} ∧ ∀ (x : X), f x ∈ Icc 0 1
s t : Set X
hs : IsClosed[inst✝] s
ht : IsClosed[inst✝] t
hst : Disjoint s t
f : C(X, ℝ)
hf : s = ⇑f ⁻¹' {0}
hfr : ∀ (x : X), f x ∈ Icc 0 1
g : C(X, ℝ)
hg : t = ⇑g ⁻¹' {0}
hgr : ∀ (x : X), g x ∈ Icc 0 1
hsn : SeparatedNhds {0} {1}
hfg : ∀ (x : X), f x + g x ≠ 0
hp : s = (fun a ↦ f a / (f a + g a)) ⁻¹' {0}
this : t = (fun a ↦ f a / (f a + g a)) ⁻¹' {1}
⊢ SeparatedNhds ((fun a ↦ f a / (f a + g a)) ⁻¹' {0}) t

Tactic:
this

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
h : ∀ (s : Set X), IsClosed[inst✝] s → ∃ f, s = ⇑f ⁻¹' {0} ∧ ∀ (x : X), f x ∈ Icc 0 1
s t : Set X
hs : IsClosed[inst✝] s
ht : IsClosed[inst✝] t
hst : Disjoint s t
f : C(X, ℝ)
hf : s = ⇑f ⁻¹' {0}
hfr : ∀ (x : X), f x ∈ Icc 0 1
g : C(X, ℝ)
hg : t = ⇑g ⁻¹' {0}
hgr : ∀ (x : X), g x ∈ Icc 0 1
hsn : SeparatedNhds {0} {1}
hfg : ∀ (x : X), f x + g x ≠ 0
hp : s = (fun a ↦ f a / (f a + g a)) ⁻¹' {0}
this : t = (fun a ↦ f a / (f a + g a)) ⁻¹' {1}
⊢ SeparatedNhds ((fun a ↦ f a / (f a + g a)) ⁻¹' {0}) ((fun a ↦ f a / (f a + g a)) ⁻¹' {1})
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case h
δ : ℝ
δpos : δ > 0
ε : ℝ := 2 / 5 * δ
εpos : 0 < ε
s : (p : GHSpace) → Set p.Rep
hs : ∀ (p : GHSpace), (s p).Finite ∧ univ ⊆ ⋃ x ∈ s p, ball x ε
N : GHSpace → ℕ := fun p ↦ Nat.card ↑(s p)
E : (p : GHSpace) → ↑(s p) ≃ Fin (Nat.card ↑(s p)) := fun p ↦ Finite.equivFin ↑(s p)
F : GHSpace → (n : ℕ) × (Fin n → Fin n → ℤ) := fun p ↦ ⟨N p, fun a b ↦ ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋⟩
p q : GHSpace
hpq : F p = F q
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x : q.Rep
this✝ : x ∈ ⋃ y ∈ s q, ball y ε
y : q.Rep
ys : y ∈ s q
hy : x ∈ ball y ε
i : ℕ := ↑((E q) ⟨y, ys⟩)
hi : ↑((E q) ⟨y, ys⟩) < Nat.card ↑(s q) := Fin.is_lt ((E q) ⟨y, ys⟩)
ihi_eq : ⟨i, hi⟩ = (E q) ⟨y, ys⟩
hiq : i < N q
hip : i < N p
z : ↑(s p) := (E p).symm ⟨i, hip⟩
C1 : (E p) z = ⟨i, hip⟩
C2 : Fin.cast Npq ⟨i, hip⟩ = ⟨i, hi⟩
C3 : (E q).symm ⟨i, hi⟩ = ⟨y, ys⟩
this : Φ z = y
⊢ dist x (Φ z) ≤ ε

Tactic:
this

Goal state after:
case h
δ : ℝ
δpos : δ > 0
ε : ℝ := 2 / 5 * δ
εpos : 0 < ε
s : (p : GHSpace) → Set p.Rep
hs : ∀ (p : GHSpace), (s p).Finite ∧ univ ⊆ ⋃ x ∈ s p, ball x ε
N : GHSpace → ℕ := fun p ↦ Nat.card ↑(s p)
E : (p : GHSpace) → ↑(s p) ≃ Fin (Nat.card ↑(s p)) := fun p ↦ Finite.equivFin ↑(s p)
F : GHSpace → (n : ℕ) × (Fin n → Fin n → ℤ) := fun p ↦ ⟨N p, fun a b ↦ ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋⟩
p q : GHSpace
hpq : F p = F q
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x : q.Rep
this✝ : x ∈ ⋃ y ∈ s q, ball y ε
y : q.Rep
ys : y ∈ s q
hy : x ∈ ball y ε
i : ℕ := ↑((E q) ⟨y, ys⟩)
hi : ↑((E q) ⟨y, ys⟩) < Nat.card ↑(s q) := Fin.is_lt ((E q) ⟨y, ys⟩)
ihi_eq : ⟨i, hi⟩ = (E q) ⟨y, ys⟩
hiq : i < N q
hip : i < N p
z : ↑(s p) := (E p).symm ⟨i, hip⟩
C1 : (E p) z = ⟨i, hip⟩
C2 : Fin.cast Npq ⟨i, hip⟩ = ⟨i, hi⟩
C3 : (E q).symm ⟨i, hi⟩ = ⟨y, ys⟩
this : Φ z = y
⊢ dist x y ≤ ε
Module: Mathlib.Topology.Sheaves.Abelian

Goal state before:
pp: case mpr
C : Type v
inst✝⁶ : Category.{u, v} C
inst✝⁵ : HasColimits C
inst✝⁴ : HasLimits C
FC : C → C → Type u_1
CC : C → Type u
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝² : PreservesFilteredColimits (CategoryTheory.forget C)
inst✝¹ : PreservesLimits (CategoryTheory.forget C)
inst✝ : Abelian C
X : TopCat
S : ShortComplex (Sheaf C X)
h : ∀ (x : ↑X), (S.map (forget C X ⋙ stalkFunctor C x)).Exact
⊢ S.Exact

Tactic:
[

Goal state after:
case mpr
C : Type v
inst✝⁶ : Category.{u, v} C
inst✝⁵ : HasColimits C
inst✝⁴ : HasLimits C
FC : C → C → Type u_1
CC : C → Type u
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝² : PreservesFilteredColimits (CategoryTheory.forget C)
inst✝¹ : PreservesLimits (CategoryTheory.forget C)
inst✝ : Abelian C
X : TopCat
S : ShortComplex (Sheaf C X)
h : ∀ (x : ↑X), (S.map (forget C X ⋙ stalkFunctor C x)).Exact
⊢ S.Exact
Module: Mathlib.Topology.Sheaves.AddCommGrpCat

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex ((Opens ↑X)ᵒᵖ ⥤ AddCommGrpCat)
hS : S.Exact
s : ↑(S.X₂.obj (op U))
h : (ConcreteCategory.hom (S.g.app (op U))) s = 0
F : ((Opens ↑X)ᵒᵖ ⥤ AddCommGrpCat) ⥤ AddCommGrpCat := (evaluation (Opens ↑X)ᵒᵖ AddCommGrpCat).obj (op U)
⊢ ∃ t, (ConcreteCategory.hom (S.f.app (op U))) t = s

Tactic:
exact (ShortComplex.ab_exact_iff (S.map F)).mp (((Functor.exact_tfae F).out 1 3 rfl rfl).mpr
    ⟨inferInstance, inferInstance⟩ S hS) _ h

Goal state after:
Module: Mathlib.Topology.Sheaves.EtaleSpace

Goal state before:
pp: X : TopCat
C : Type u
inst✝⁴ : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
inst✝¹ : Limits.HasColimits C
F : Presheaf C X
inst✝ : Limits.PreservesFilteredColimits (forget C)
U : Opens ↑X
hF_bij : ∀ (x : ↑X) (hx : x ∈ U), Bijective ⇑(ConcreteCategory.hom (F.germ U x hx))
x : ↑X
hx : x ∈ U
⊢ Continuous fun s ↦ WithTopology.toTopology ⊥ ((ConcreteCategory.hom (F.germ U x hx)) (surjInv ⋯ (↑s).germ))

Tactic:
[

Goal state after:
X : TopCat
C : Type u
inst✝⁴ : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
inst✝¹ : Limits.HasColimits C
F : Presheaf C X
inst✝ : Limits.PreservesFilteredColimits (forget C)
U : Opens ↑X
hF_bij : ∀ (x : ↑X) (hx : x ∈ U), Bijective ⇑(ConcreteCategory.hom (F.germ U x hx))
x : ↑X
hx : x ∈ U
⊢ Continuous fun s ↦ WithTopology.toTopology ⊥ ((ConcreteCategory.hom (F.germ U x hx)) (surjInv ⋯ (↑s).germ))
Module: Mathlib.Topology.Sheaves.EtaleSpace

Goal state before:
pp: X : TopCat
C : Type u
inst✝⁴ : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
inst✝¹ : Limits.HasColimits C
F : Presheaf C X
inst✝ : Limits.PreservesFilteredColimits (forget C)
U : Opens ↑X
hF_bij : ∀ (x : ↑X) (hx : x ∈ U), Bijective ⇑(ConcreteCategory.hom (F.germ U x hx))
x : ↑X
hx : x ∈ U
⊢ Continuous fun x_1 ↦
    ⟨{ base := ↑x_1.1, germ := (ConcreteCategory.hom (F.germ U ↑x_1.1 ⋯)) (surjInv ⋯ x_1.2.ofTopology) }, ⋯⟩

Tactic:
[

Goal state after:
X : TopCat
C : Type u
inst✝⁴ : Category.{v, u} C
CC : C → Type v
FC : C → C → Type w
inst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory C FC
inst✝¹ : Limits.HasColimits C
F : Presheaf C X
inst✝ : Limits.PreservesFilteredColimits (forget C)
U : Opens ↑X
hF_bij : ∀ (x : ↑X) (hx : x ∈ U), Bijective ⇑(ConcreteCategory.hom (F.germ U x hx))
x : ↑X
hx : x ∈ U
⊢ Continuous fun x_1 ↦
    ⟨{ base := ↑x_1.1, germ := (ConcreteCategory.hom (F.germ U ↑x_1.1 ⋯)) (surjInv ⋯ x_1.2.ofTopology) }, ⋯⟩
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:

Tactic:
|dist x y - dist (Ψ x) (Ψ y)|
        _ = ε * (ε⁻¹ * |dist x y - dist (Ψ x) (Ψ y)|) := by
            #adaptation_note /-- Before https://github.com/leanprover/lean4/pull/13166
            (replacing grind's canonicalizer with a type-directed normalizer), `grind` closed
            this goal. It is not yet clear whether this is due to defeq abuse in Mathlib or a
            problem in the new canonicalizer; a minimization would help. The original proof was:
            `grind` -/
            field_simp
        _ ≤ ε * 1 := by gcongr
        _ = ε := mul_one _

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case h
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x : q.Rep
this✝ : x ∈ ⋃ y ∈ s q, ball y (u n)
y : q.Rep
ys : y ∈ s q
hy : x ∈ ball y (u n)
i : ℕ := ↑((E q) ⟨y, ys⟩)
hi : ↑((E q) ⟨y, ys⟩) < N q := ((E q) ⟨y, ys⟩).isLt
ihi_eq : ⟨i, hi⟩ = (E q) ⟨y, ys⟩
hiq : i < N q
hip : i < N p
z : ↑(s p) := (E p).symm ⟨i, hip⟩
C1 : (E p) z = ⟨i, hip⟩
C2 : Fin.cast Npq ⟨i, hip⟩ = ⟨i, hi⟩
C3 : (E q).symm ⟨i, hi⟩ = ⟨y, ys⟩
this : Φ z = y
⊢ dist x (Φ z) ≤ ε

Tactic:
this

Goal state after:
case h
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x : q.Rep
this✝ : x ∈ ⋃ y ∈ s q, ball y (u n)
y : q.Rep
ys : y ∈ s q
hy : x ∈ ball y (u n)
i : ℕ := ↑((E q) ⟨y, ys⟩)
hi : ↑((E q) ⟨y, ys⟩) < N q := ((E q) ⟨y, ys⟩).isLt
ihi_eq : ⟨i, hi⟩ = (E q) ⟨y, ys⟩
hiq : i < N q
hip : i < N p
z : ↑(s p) := (E p).symm ⟨i, hip⟩
C1 : (E p) z = ⟨i, hip⟩
C2 : Fin.cast Npq ⟨i, hip⟩ = ⟨i, hi⟩
C3 : (E q).symm ⟨i, hi⟩ = ⟨y, ys⟩
this : Φ z = y
⊢ dist x y ≤ ε
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case refine_3
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
this : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
⊢ |dist x y - dist (Φ x) (Φ y)| ≤ ε

Tactic:
this

Goal state after:
case refine_3
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p : GHSpace) → Set p.Rep
N : GHSpace → ℕ
hN : ∀ (p : GHSpace), N p ≤ K n
E : (p : GHSpace) → ↑(s p) ≃ Fin (N p)
hs : ∀ p ∈ t, univ ⊆ ⋃ x ∈ s p, ball x (u n)
M : ℕ := ⌊ε⁻¹ * max C 0⌋₊
F : GHSpace → (k : Fin (K n).succ) × (Fin ↑k → Fin ↑k → Fin M.succ) :=
  fun p ↦ ⟨⟨N p, ⋯⟩, fun a b ↦ ⟨min M ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋₊, ⋯⟩⟩
p : GHSpace
pt : p ∈ t
q : GHSpace
qt : q ∈ t
hpq : (fun p ↦ F ↑p) ⟨p, pt⟩ = (fun p ↦ F ↑p) ⟨q, qt⟩
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
this : dist (Φ x) (Φ y) = dist (Ψ x) (Ψ y)
⊢ |dist x y - dist (Ψ x) (Ψ y)| ≤ ε
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : MetricSpace Y
inst✝³ : CompactSpace X
inst✝² : CompactSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
⊢ (fun p ↦ dist p.1 p.2) ∈ candidates X Y

Tactic:
[

Goal state after:
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : MetricSpace Y
inst✝³ : CompactSpace X
inst✝² : CompactSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
⊢ (fun p ↦ dist p.1 p.2) ∈ candidates X Y
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ ∀ (j : WalkingParallelPair),
    𝟙 ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).pt ≫
        ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app j =
      ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app j

Tactic:
rintro ⟨_ | _⟩
              · dsimp
                ext
                simp
              · dsimp
                ext
                simp

Goal state after:
Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone (SheafConditionEqualizerProducts.diagram F U)
⊢ ∀ (j : WalkingParallelPair),
    𝟙 ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).pt ≫
        ((coneEquivInverse F U ⋙ coneEquivFunctor F U).obj c).π.app j =
      ((𝟭 (Cone (SheafConditionEqualizerProducts.diagram F U))).obj c).π.app j

Tactic:
rintro ⟨_ | _⟩
              · dsimp
                ext
                simp
              · dsimp
                ext
                simp

Goal state after: