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

Goal state before:
pp: 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)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
⊢ ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ↑((F p).snd ((E p) x) ((E p) y))

Tactic:
by

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

Goal state before:
pp: 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)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
⊢ ↑((F p).snd ((E p) x) ((E p) y)) = min M ⌊ε⁻¹ * dist x y⌋₊

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ x, ⨅ y, f (inl x, inr y) ≤ ⨆ x, ⨅ y, g (inl x, inr y) + dist f g
⊢ ∀ (x : X), (⨅ y, g (inl x, inr y)) + dist f g = ⨅ y, g (inl x, inr y) + dist f g

Tactic:
by

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

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
s : ↑(S.X₃.obj.obj (op U))
t : Under S.g s
ht : ∀ (a : Under S.g s), Nonempty (a ⟶ t) → Nonempty (t ⟶ a)
tle : unop (StructuredArrow.right t).fst ≤ U
tcomp :
  (s |_ unop (StructuredArrow.right t).fst) tle =
    (ConcreteCategory.hom (S.g.hom.app (StructuredArrow.right t).fst)) (StructuredArrow.right t).snd
x : ↑X
hx : x ∈ U
this✝¹ : Presheaf.IsLocallySurjective S.g.hom
W : Opens ↑X
Wle : W ≤ U
hW : x ∈ W
t₁ : ↑(S.X₂.obj.obj (op W))
ht₁ : (ConcreteCategory.hom (S.g.hom.app (op W))) t₁ = (s |_ W) Wle
t₂ : ToType (S.X₂.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W))) :=
  ((StructuredArrow.right t).snd |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ -
    (t₁ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
this✝ : (ConcreteCategory.hom (S.g.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₂ = 0
t₃ : ↑(S.X₁.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W)))
ht₃ : (ConcreteCategory.hom (S.f.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₃ = t₂
t₄ : ↑(S.X₁.obj.obj (op W))
ht₄ : (t₄ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = t₃
f : Fin 2 → Opens ↑X := ![unop (StructuredArrow.right t).fst, W]
sf : (i : Fin 2) → ↑(S.X₂.obj.obj (op (f i))) :=
  fun x ↦
    match x with
    | 0 => (StructuredArrow.right t).snd
    | 1 => t₁ + (ConcreteCategory.hom (S.f.hom.app (op W))) t₄
this : (sf 0 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = (sf 1 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
t₅ : ToType (S.X₂.obj.obj (op (iSup f)))
ht₅ : IsGluing S.X₂.obj f sf t₅
right✝ : ∀ (y : ToType (S.X₂.obj.obj (op (iSup f)))), (fun t₅ ↦ IsGluing S.X₂.obj f sf t₅) y → y = t₅
le : iSup f ≤ U
⊢ ∀ (i : Fin 2), f i ≤ U

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ x, ⨅ y, f (inl x, inr y) ≤ ⨆ x, ⨅ y, g (inl x, inr y) + dist f g
E1 : ∀ (x : X), (⨅ y, g (inl x, inr y)) + dist f g = ⨅ y, g (inl x, inr y) + dist f g
⊢ (⨆ x, ⨅ y, g (inl x, inr y)) + dist f g = ⨆ x, (⨅ y, g (inl x, inr y)) + dist f g

Tactic:
by

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

Goal state before:
pp: 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)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
⊢ min M ⌊ε⁻¹ * dist x y⌋₊ = ⌊ε⁻¹ * dist x y⌋₊

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
⊢ ⨆ x, ⨅ y, f (inl x, inr y) ≤ (⨆ x, ⨅ y, g (inl x, inr y)) + dist f g

Tactic:
by

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

Goal state before:
pp: 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)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
⊢ ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩) = ↑((F q).snd ((E q) (Ψ x)) ((E q) (Ψ y)))

Tactic:
by

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

Goal state before:
pp: 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)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
⊢ ↑((F q).snd ((E q) (Ψ x)) ((E q) (Ψ y))) = min M ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ y, ⨅ x, f (inl x, inr y) ≤ ⨆ y, ⨅ x, g (inl x, inr y) + dist f g
⊢ ∀ (y : Y), (⨅ x, g (inl x, inr y)) + dist f g = ⨅ x, g (inl x, inr y) + dist f g

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ y, ⨅ x, f (inl x, inr y) ≤ ⨆ y, ⨅ x, g (inl x, inr y) + dist f g
E1 : ∀ (y : Y), (⨅ x, g (inl x, inr y)) + dist f g = ⨅ x, g (inl x, inr y) + dist f g
⊢ (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g = ⨆ y, (⨅ x, g (inl x, inr y)) + dist f g

Tactic:
by

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

Goal state before:
pp: 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)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
⊢ min M ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊ = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊

Tactic:
refine min_eq_right (Nat.floor_mono ?_)
            refine mul_le_mul_of_nonneg_left (le_trans ?_ (le_max_left _ _)) (inv_pos.2 εpos).le
            change dist (Ψ x : q.Rep) (Ψ y) ≤ C
            refine (dist_le_diam_of_mem isCompact_univ.isBounded (mem_univ _) (mem_univ _)).trans ?_
            exact hdiam q qt

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
⊢ ⨆ y, ⨅ x, f (inl x, inr y) ≤ (⨆ y, ⨅ x, g (inl x, inr y)) + dist f g

Tactic:
by

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

Goal state before:
pp: 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)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
⊢ min M ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊ = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
⊢ ⨆ x, ⨅ y, f (inl x, inr y) ≤ HD g + dist f g

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
⊢ ⨆ y, ⨅ x, f (inl x, inr y) ≤ HD g + dist f g

Tactic:
by

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)
⊢ ∀ ⦃X_1 Y : (CategoryTheory.Pairwise ι)ᵒᵖ⦄ (f : X_1 ⟶ Y),
    ((Functor.const (CategoryTheory.Pairwise ι)ᵒᵖ).obj c.pt).map f ≫
        Opposite.rec
          (fun x ↦
            Pairwise.casesOn x (fun i ↦ c.π.app WalkingParallelPair.zero ≫ Pi.π (fun i ↦ F.obj (op (U i))) i) fun i j ↦
              c.π.app WalkingParallelPair.one ≫ Pi.π (fun p ↦ F.obj (op (U p.1 ⊓ U p.2))) (i, j))
          Y =
      Opposite.rec
          (fun x ↦
            Pairwise.casesOn x (fun i ↦ c.π.app WalkingParallelPair.zero ≫ Pi.π (fun i ↦ F.obj (op (U i))) i) fun i j ↦
              c.π.app WalkingParallelPair.one ≫ Pi.π (fun p ↦ F.obj (op (U p.1 ⊓ U p.2))) (i, j))
          X_1 ≫
        ((diagram U).op ⋙ F).map f

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace X
inst✝ : CompactSpace Y
⊢ IsCompact (candidatesB X Y)

Tactic:
by

Goal state after:
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:
by

Goal state after:
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
x : X
⊢ BddBelow (range fun y ↦ (candidatesBDist X Y) (inl x, inr y))

Tactic:
by

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 c' : Cone (SheafConditionEqualizerProducts.diagram F U)
f : c ⟶ c'
⊢ ∀ (j : (CategoryTheory.Pairwise ι)ᵒᵖ),
    f.hom ≫ (coneEquivInverseObj F U c').π.app j = (coneEquivInverseObj F U c).π.app j

Tactic:
by

Goal state after:
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
x : X
A : ⨅ y, (candidatesBDist X Y) (inl x, inr y) ≤ (candidatesBDist X Y) (inl x, inr default)
⊢ dist x default + 1 + dist default default ≤ diam univ + 1 + diam univ

Tactic:
by

Goal state after:
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
y : Y
⊢ BddBelow (range fun x ↦ (candidatesBDist X Y) (inl x, inr y))

Tactic:
by

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

Goal state before:
pp: 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)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
Ap : ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ⌊ε⁻¹ * dist x y⌋₊
Aq : ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩) = ⌊ε⁻¹ * dist (Ψ x) (Ψ y)⌋₊
⊢ ↑((F p).snd ⟨i, hip⟩ ⟨j, hjp⟩) = ↑((F q).snd ⟨i, hiq⟩ ⟨j, hjq⟩)

Tactic:
by

Goal state after:
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
y : Y
A : ⨅ x, (candidatesBDist X Y) (inl x, inr y) ≤ (candidatesBDist X Y) (inl default, inr y)
⊢ dist default default + 1 + dist default y ≤ diam univ + 1 + diam univ

Tactic:
by

Goal state after:
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
⊢ HD (candidatesBDist X Y) ≤ diam univ + 1 + diam univ

Tactic:
by

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 ((diagram U).op ⋙ F)
j : (CategoryTheory.Pairwise ι)ᵒᵖ
⊢ 𝟙 ((𝟭 (Cone ((diagram U).op ⋙ F))).obj c).pt ≫ ((coneEquivFunctor F U ⋙ coneEquivInverse F U).obj c).π.app j =
    ((𝟭 (Cone ((diagram U).op ⋙ F))).obj c).π.app j

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
⊢ optimalGHDist X Y ∈ candidatesB X Y

Tactic:
by

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

Goal state before:
pp: X : TopCat
U : Opens ↑X
S : ShortComplex (Sheaf AddCommGrpCat X)
hS : S.ShortExact
inst✝ : S.X₁.IsFlasque
s : ↑(S.X₃.obj.obj (op U))
t : Under S.g s
ht : ∀ (a : Under S.g s), Nonempty (a ⟶ t) → Nonempty (t ⟶ a)
tle : unop (StructuredArrow.right t).fst ≤ U
tcomp :
  (s |_ unop (StructuredArrow.right t).fst) tle =
    (ConcreteCategory.hom (S.g.hom.app (StructuredArrow.right t).fst)) (StructuredArrow.right t).snd
x : ↑X
hx : x ∈ U
this✝¹ : Presheaf.IsLocallySurjective S.g.hom
W : Opens ↑X
Wle : W ≤ U
hW : x ∈ W
t₁ : ↑(S.X₂.obj.obj (op W))
ht₁ : (ConcreteCategory.hom (S.g.hom.app (op W))) t₁ = (s |_ W) Wle
t₂ : ToType (S.X₂.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W))) :=
  ((StructuredArrow.right t).snd |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ -
    (t₁ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
this✝ : (ConcreteCategory.hom (S.g.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₂ = 0
t₃ : ↑(S.X₁.obj.obj (op (unop (StructuredArrow.right t).fst ⊓ W)))
ht₃ : (ConcreteCategory.hom (S.f.hom.app (op (unop (StructuredArrow.right t).fst ⊓ W)))) t₃ = t₂
t₄ : ↑(S.X₁.obj.obj (op W))
ht₄ : (t₄ |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = t₃
f : Fin 2 → Opens ↑X := ![unop (StructuredArrow.right t).fst, W]
sf : (i : Fin 2) → ↑(S.X₂.obj.obj (op (f i))) :=
  fun x ↦
    match x with
    | 0 => (StructuredArrow.right t).snd
    | 1 => t₁ + (ConcreteCategory.hom (S.f.hom.app (op W))) t₄
this : (sf 0 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯ = (sf 1 |_ (unop (StructuredArrow.right t).fst ⊓ W)) ⋯
t₅ : ToType (S.X₂.obj.obj (op (iSup f)))
ht₅ : IsGluing S.X₂.obj f sf t₅
right✝ : ∀ (y : ToType (S.X₂.obj.obj (op (iSup f)))), (fun t₅ ↦ IsGluing S.X₂.obj f sf t₅) y → y = t₅
le : iSup f ≤ U
⊢ S.f.hom.app (op W) ≫ S.g.hom.app (op W) = 0

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
⊢ IsCompact univ

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : Cb X Y
h : f ∈ candidatesB X Y
r : ℝ
hr : HD (optimalGHDist X Y) < r
z : X
I1 : ⨆ x, ⨅ y, (optimalGHDist X Y) (inl x, inr y) < r
⊢ BddAbove (range fun x ↦ ⨅ y, (optimalGHDist X Y) (inl x, inr y))

Tactic:
by

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

Goal state before:
pp: X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : Cb X Y
h : f ∈ candidatesB X Y
r : ℝ
hr : HD (optimalGHDist X Y) < r
⊢ ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r

Tactic:
by

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 ((diagram U).op ⋙ F)
j : (CategoryTheory.Pairwise ι)ᵒᵖ
⊢ 𝟙 ((coneEquivFunctor F U ⋙ coneEquivInverse F U).obj c).pt ≫ ((𝟭 (Cone ((diagram U).op ⋙ F))).obj c).π.app j =
    ((coneEquivFunctor F U ⋙ coneEquivInverse F U).obj c).π.app j

Tactic:
by

Goal state after: