mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Convenient.GeneratedBy

Goal state before:
pp: ι : Type t
X : ι → Type u
inst✝¹ : (i : ι) → TopologicalSpace (X i)
Y : Type v
tY : TopologicalSpace Y
Z : Type v'
inst✝ : TopologicalSpace Z
U : Set (WithGeneratedByTopology X (WithGeneratedByTopology X Y))
hU :
  ∀ ⦃i : ι⦄ (f : C(X i, WithGeneratedByTopology X Y)),
    IsOpen[inst✝¹ i] (⇑f ⁻¹' ⇑WithGeneratedByTopology.equiv.symm ⁻¹' U)
i : ι
f : C(X i, Y)
⊢ IsOpen[inst✝¹ i] (⇑f ⁻¹' ⇑WithGeneratedByTopology.equiv.symm ⁻¹' ⇑WithGeneratedByTopology.equiv.symm ⁻¹' U)

Tactic:
refine hU ⟨WithGeneratedByTopology.equiv.symm ∘ f, ?_⟩

Goal state after:
ι : Type t
X : ι → Type u
inst✝¹ : (i : ι) → TopologicalSpace (X i)
Y : Type v
tY : TopologicalSpace Y
Z : Type v'
inst✝ : TopologicalSpace Z
U : Set (WithGeneratedByTopology X (WithGeneratedByTopology X Y))
hU :
  ∀ ⦃i : ι⦄ (f : C(X i, WithGeneratedByTopology X Y)),
    IsOpen[inst✝¹ i] (⇑f ⁻¹' ⇑WithGeneratedByTopology.equiv.symm ⁻¹' U)
i : ι
f : C(X i, Y)
⊢ Continuous[inst✝¹ i, _] (⇑WithGeneratedByTopology.equiv.symm ∘ ⇑f)
Module: Mathlib.Topology.CWComplex.Classical.Basic

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
n : ℕ
i : cell C n
⊢ cellFrontier n i ∪ openCell n i = closedCell n i

Tactic:
cellFrontier,

Goal state after:
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝ : RelCWComplex C D
n : ℕ
i : cell C n
⊢ ↑(map n i) '' sphere 0 1 ∪ openCell n i = closedCell n i
Module: Mathlib.Topology.CWComplex.Classical.Basic

Goal state before:
pp: X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : RelCWComplex C D
inst✝ : T2Space X
A : Set X
n : ℕ
hn : ∀ m ≤ n, ∀ (j : cell C m), IsClosed[t] (A ∩ closedCell m j)
j : cell C (n + 1)
hD : IsClosed[t] (A ∩ D)
I : (m : ℕ) → Finset (cell C m)
hI : cellFrontier (n + 1) j ⊆ D ∪ ⋃ m, ⋃ (_ : m < n + 1), ⋃ j ∈ I m, closedCell m j
⊢ IsClosed[t] (A ∩ ((D ∪ ⋃ m, ⋃ (_ : m < n + 1), ⋃ j ∈ I m, closedCell m j) ∩ cellFrontier (n + 1) j))

Tactic:
← inter_assoc

Goal state after:
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : RelCWComplex C D
inst✝ : T2Space X
A : Set X
n : ℕ
hn : ∀ m ≤ n, ∀ (j : cell C m), IsClosed[t] (A ∩ closedCell m j)
j : cell C (n + 1)
hD : IsClosed[t] (A ∩ D)
I : (m : ℕ) → Finset (cell C m)
hI : cellFrontier (n + 1) j ⊆ D ∪ ⋃ m, ⋃ (_ : m < n + 1), ⋃ j ∈ I m, closedCell m j
⊢ IsClosed[t] (A ∩ (D ∪ ⋃ m, ⋃ (_ : m < n + 1), ⋃ j ∈ I m, closedCell m j) ∩ cellFrontier (n + 1) j)
Module: Mathlib.Topology.CWComplex.Classical.Basic

Goal state before:
pp: case hi.inr
X : Type u_1
t : TopologicalSpace X
C D : Set X
inst✝¹ : RelCWComplex C D
inst✝ : T2Space X
A : Set X
hAC : A ⊆ C
hDA : IsClosed[t] (A ∩ D)
n : ℕ
hn : ∀ m ≤ n, ∀ (j : cell C m), IsClosed[t] (A ∩ closedCell m j)
j : cell C (n + 1)
h2 : IsClosed[t] (A ∩ closedCell n.succ j)
⊢ IsClosed[t] (A ∩ closedCell (n + 1) j)

Tactic:
· exact h2

Goal state after:
Module: Mathlib.Topology.Category.Profinite.Nobeling.Basic

Goal state before:
pp: I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
l : Products I
J : I → Prop
inst✝ : (j : I) → Decidable (J j)
h : isGood (π C J) l
i : I
hi : i ∈ ↑l
h' : ¬J i
w✝ : I → Bool
left✝ : w✝ ∈ C
⊢ (if ∀ i ∈ ↑l, ↑⟨Proj J w✝, ⋯⟩ i = true then 1 else 0) = 0 ⟨Proj J w✝, ⋯⟩

Tactic:
if_neg fun h ↦ ?_,

Goal state after:
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
l : Products I
J : I → Prop
inst✝ : (j : I) → Decidable (J j)
h : isGood (π C J) l
i : I
hi : i ∈ ↑l
h' : ¬J i
w✝ : I → Bool
left✝ : w✝ ∈ C
⊢ 0 = 0 ⟨Proj J w✝, ⋯⟩

I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
l : Products I
J : I → Prop
inst✝ : (j : I) → Decidable (J j)
h✝ : isGood (π C J) l
i : I
hi : i ∈ ↑l
h' : ¬J i
w✝ : I → Bool
left✝ : w✝ ∈ C
h : ∀ i ∈ ↑l, ↑⟨Proj J w✝, ⋯⟩ i = true
⊢ False
Module: Mathlib.Topology.Category.Profinite.Nobeling.ZeroLimit

Goal state before:
pp: case refine_1
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
a b : ↑(range (π C fun x ↦ ord I x < o))
hab : (fun x ↦ ⟨(πs C o) ↑x, ⋯⟩) a = (fun x ↦ ⟨(πs C o) ↑x, ⋯⟩) b
⊢ ↑a = ↑b

Tactic:
simp only [Subtype.mk.injEq] at hab

Goal state after:
case refine_1
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
a b : ↑(range (π C fun x ↦ ord I x < o))
hab : (πs C o) ↑a = (πs C o) ↑b
⊢ ↑a = ↑b
Module: Mathlib.Topology.Category.Profinite.Nobeling.Induction

Goal state before:
pp: I : Type u
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
ho : Order.IsSuccLimit o
⊢ (∀ o' < o, P I o') → P I o

Tactic:
by
  intro h hho C hC hsC
  rw [linearIndependent_iff_union_smaller C ho hsC, linearIndependent_subtype_iff]
  exact linearIndepOn_iUnion_of_directed
    (Monotone.directed_le fun _ _ h ↦ GoodProducts.smaller_mono C h) fun ⟨o', ho'⟩ ↦
    (linearIndependent_iff_smaller _ _).mp (h o' ho' (ho'.le.trans hho)
    (π C (ord I · < o')) (isClosed_proj _ _ hC) (contained_proj _ _))

Goal state after:
Module: Mathlib.Topology.Category.Profinite.Nobeling.Successor

Goal state before:
pp: I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC✝ : contained C (Order.succ o)
ho✝ : o < Ordinal.type fun x1 x2 ↦ x1 < x2
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
⊢ GoodProducts (π C fun x ↦ ord I x < o) ∪ MaxProducts C ho = GoodProducts C

Tactic:
union_succ C hsC ho

Goal state after:
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hC : IsClosed C
hsC✝ : contained C (Order.succ o)
ho✝ : o < Ordinal.type fun x1 x2 ↦ x1 < x2
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
⊢ GoodProducts (π C fun x ↦ ord I x < o) ∪ MaxProducts C ho = GoodProducts (π C fun x ↦ ord I x < o) ∪ MaxProducts C ho
Module: Mathlib.Topology.Category.Profinite.Nobeling.Successor

Goal state before:
pp: case neg
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 ↦ x1 < x2
l : ↑(MaxProducts C ho)
q : Products I
hq : q < (↑l).Tail
this✝ : Inhabited I
a : I
ha : a ∈ ↑q
hM : ¬↑(↑l).Tail = []
this : List.Pairwise (fun x1 x2 ↦ x1 > x2) (term I ho :: ↑(↑l).Tail)
⊢ (↑(↑l).Tail).head! < term I ho

Tactic:
exact List.rel_of_pairwise_cons this (List.head!_mem_self hM)

Goal state after:
Module: Mathlib.Topology.Compactness.CountablyCompact

Goal state before:
pp: E : Type u_2
inst✝ : TopologicalSpace E
A : Set E
h : ∀ (U : ℕ → Set E), (∀ (i : ℕ), IsOpen[inst✝] (U i)) → A ⊆ ⋃ i, U i → ∃ t, A ⊆ ⋃ i ∈ t, U i
⊢ IsCountablyCompact A

Tactic:
refine IsCountablyCompact.of_seq_clusterPt fun x hx => ?_
    by_contra! hac
    let V : ℕ → Set E := fun n => (closure (x '' Ici n))ᶜ
    have hVmono : Monotone V := fun _ _ hmn =>
      compl_subset_compl.2 <| closure_mono <| image_mono <| Ici_subset_Ici.2 hmn
    simp only [mapClusterPt_atTop_iff_forall_mem_closure, not_forall] at hac
    have hAV : A ⊆ ⋃ n, V n := fun a haA => mem_iUnion.2 (hac a haA)
    obtain ⟨t, ht⟩ := h V (fun _ => isClosed_closure.isOpen_compl) hAV
    obtain ⟨N, hN⟩ := eventually_atTop.mp hx
    let m := max N (t.sup id)
    obtain ⟨j, hjt, hjV⟩ := mem_iUnion₂.mp (ht (hN m (le_max_left _ _)))
    have hxmV : x m ∈ V m := hVmono ((Finset.le_sup hjt).trans (le_max_right _ _)) hjV
    exact hxmV (subset_closure ⟨m, mem_Ici.mpr le_rfl, rfl⟩)

Goal state after:
Module: Mathlib.Topology.ContinuousMap.SecondCountableSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : SecondCountableTopology X
inst✝¹ : LocallyCompactSpace X
inst✝ : SecondCountableTopology Y
this : ∀ (U : ↑(countableBasis X)), LocallyCompactSpace ↑↑U
K : (U : ↑(countableBasis X)) → CompactExhaustion ↑↑U := fun U ↦ CompactExhaustion.choice ↑↑U
f : C(X, Y)
V : Set Y
hVo : IsOpen[inst✝³] V
x : X
hxV : x ∈ ⇑f ⁻¹' V
⊢ ∃ U ∈ countableBasis X, x ∈ U ∧ U ⊆ ⇑f ⁻¹' V

Tactic:
rw [← (isBasis_countableBasis _).mem_nhds_iff]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : SecondCountableTopology X
inst✝¹ : LocallyCompactSpace X
inst✝ : SecondCountableTopology Y
this : ∀ (U : ↑(countableBasis X)), LocallyCompactSpace ↑↑U
K : (U : ↑(countableBasis X)) → CompactExhaustion ↑↑U := fun U ↦ CompactExhaustion.choice ↑↑U
f : C(X, Y)
V : Set Y
hVo : IsOpen[inst✝³] V
x : X
hxV : x ∈ ⇑f ⁻¹' V
⊢ ⇑f ⁻¹' V ∈ 𝓝 x
Module: Mathlib.Topology.Homotopy.HomotopyGroup

Goal state before:
pp: case mpr.inl
N : Type u_1
M : Type u_3
y : M ⊕ N → ↑I
m : M
hm : (y ∘ Sum.inl) m = 0 ∨ (y ∘ Sum.inl) m = 1
⊢ y ∈ Cube.boundary (M ⊕ N)

Tactic:
exact ⟨Sum.inl m, hm⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s : Set α
inst✝ : CompleteSpace α
⊢ CompleteSpace (Closeds α)

Tactic:
let B : ℕ → ℝ≥0∞ := fun n => 2⁻¹ ^ n

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : EMetricSpace α
inst✝¹ : EMetricSpace β
s : Set α
inst✝ : CompleteSpace α
B : ℕ → ℝ≥0∞ := fun n ↦ 2⁻¹ ^ n
⊢ CompleteSpace (Closeds α)
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: case hbc
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
f : ProdSpaceFun X Y
x y z t : X ⊕ Y
fA : f ∈ candidates X Y
⊢ dist x z ≤ max (dist x z) (dist t y)

Tactic:
apply le_max_left

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

Goal state before:
pp: X : Type u_1
inst✝² : Nonempty X
inst✝¹ : MetricSpace X
inst✝ : CompactSpace X
this : TopologicalSpace.SeparableSpace X := TopologicalSpace.SecondCountableTopology.to_separableSpace
emb : X → ℕ → ↑unitInterval
h_emb : Topology.IsEmbedding emb
KH : Set (ℕ → ↑unitInterval) := Set.range emb
g : X ≃ₜ ↑KH := h_emb.toHomeomorph
KC : Set (ℕ → Bool) := cantorToHilbert ⁻¹' KH
hKC_closed : IsClosed KC
f : (ℕ → Bool) → ℕ → Bool
hf_continuous : Continuous f
hf_surjective : Set.range f = KC
f' : (ℕ → Bool) → ↑KC := Subtype.coind f ⋯
⊢ Set.SurjOn f Set.univ KC

Tactic:
grind [Set.SurjOn]

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

Goal state before:
pp: case neg
X : Type u_1
Y : Type u_2
inst✝¹ : MetricSpace X
inst✝ : EMetricSpace Y
r : ℝ≥0
f : X → Y
hf : MemHolder r f
x₁ x₂ : X
hx : ¬x₁ = x₂
h₁ : edist x₁ x₂ ^ ↑r ≠ 0
h₂ : edist x₁ x₂ ^ ↑r ≠ ∞
⊢ edist (f x₁) (f x₂) / edist x₁ x₂ ^ ↑r ≤ ⨅ C, ⨅ (_ : HolderWith C r f), ↑C

Tactic:
refine le_iInf₂ fun C hC => ?_

Goal state after:
case neg
X : Type u_1
Y : Type u_2
inst✝¹ : MetricSpace X
inst✝ : EMetricSpace Y
r : ℝ≥0
f : X → Y
hf : MemHolder r f
x₁ x₂ : X
hx : ¬x₁ = x₂
h₁ : edist x₁ x₂ ^ ↑r ≠ 0
h₂ : edist x₁ x₂ ^ ↑r ≠ ∞
C : ℝ≥0
hC : HolderWith C r f
⊢ edist (f x₁) (f x₂) / edist x₁ x₂ ^ ↑r ≤ ↑C
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: α : Type u_1
inst✝¹ : EMetricSpace α
s : Set α
inst✝ : Finite ↑s
⊢ 0 < s.einfsep

Tactic:
exact einfsep_pos_of_finite

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

Goal state before:
pp: X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoEMetricSpace X
x : Snowflaking X α hα₀ hα₁
d : ℝ≥0∞
⊢ ⇑toSnowflaking ⁻¹' eball x d = eball (ofSnowflaking x) (d ^ α⁻¹)

Tactic:
rw [toSnowflaking.preimage_eq_iff_eq_image, image_toSnowflaking_eball,
    toSnowflaking_ofSnowflaking, ENNReal.rpow_inv_rpow hα₀.ne']

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

Goal state before:
pp: X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoMetricSpace X
⊢ ∀ (x y : Snowflaking X α hα₀ hα₁), 0 ≤ dist x y

Tactic:
intro x y

Goal state after:
X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoMetricSpace X
x y : Snowflaking X α hα₀ hα₁
⊢ 0 ≤ dist x y
Module: Mathlib.Topology.MetricSpace.Snowflaking

Goal state before:
pp: X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoMetricSpace X
⊢ ∀ (x y : Snowflaking X α hα₀ hα₁), edist x y = ENNReal.ofReal (dist x y)

Tactic:
intro x y

Goal state after:
X : Type u_1
α : ℝ
hα₀ : 0 < α
hα₁ : α ≤ 1
inst✝ : PseudoMetricSpace X
x y : Snowflaking X α hα₀ hα₁
⊢ edist x y = ENNReal.ofReal (dist x y)
Module: Mathlib.Topology.NatEmbedding

Goal state before:
pp: case neg.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : Infinite X
x : X
hx : (𝓝[≠] x).NeBot
this✝ : Std.Symm Disjoint
S : Finset (Set X)
hS : ∀ x_1 ∈ S, x_1.Nonempty ∧ IsOpen[inst✝²] x_1 ∧ x_1ᶜ ∈ 𝓝 x
this : (⋂ U ∈ S, interior Uᶜ) \ {x} ∈ 𝓝[≠] x
y : X
hyU : y ∈ ⋂ U ∈ S, interior Uᶜ
hyx : y ≠ x
V W : Set X
hVo : IsOpen[inst✝²] V
hWo : IsOpen[inst✝²] W
hyV : y ∈ V
hxW : x ∈ W
hVW : Disjoint V W
⊢ (V ∩ ⋂ U ∈ S, interior Uᶜ)ᶜ ∈ 𝓝 x

Tactic:
refine mem_of_superset (hWo.mem_nhds hxW) fun z hzW ⟨hzV, _⟩ ↦ ?_
      exact disjoint_left.1 hVW hzV hzW

Goal state after:
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
⊢ ∀ (s : Set β), IsOpen[inst✝²] s → IsOpen[inst✝³] (f ⁻¹' s)

Tactic:
isOpen_iff_isUpperSet

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
⊢ ∀ (s : Set β), IsUpperSet s → IsUpperSet (f ⁻¹' s)
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
⊢ ∀ (s : Set β), IsOpen[inst✝²] s → IsOpen[inst✝³] (f ⁻¹' s)

Tactic:
isOpen_iff_isUpperSet

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
⊢ ∀ (s : Set β), IsOpen[inst✝²] s → IsUpperSet (f ⁻¹' s)
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ x) (Φ y)| ≤ ε₂
δ : ℝ
δ0 : 0 < δ
xX : X
h✝ : xX ∈ univ
⊢ ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃ + δ

Tactic:
rcases hs xX with ⟨xs, hxs, Dxs⟩

Goal state after:
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ x) (Φ y)| ≤ ε₂
δ : ℝ
δ0 : 0 < δ
xX : X
h✝ : xX ∈ univ
xs : X
hxs : xs ∈ s
Dxs : dist xX xs ≤ ε₁
⊢ ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃ + δ
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ x) (Φ y)| ≤ ε₂
δ : ℝ
δ0 : 0 < δ
xX : X
h✝ : xX ∈ univ
xs : X
hxs : xs ∈ s
Dxs : dist xX xs ≤ ε₁
sne : s.Nonempty
x✝¹ : Nonempty ↑s := Nonempty.to_subtype sne
this✝⁴ : 0 ≤ ε₂
this✝³ : ∀ (p q : ↑s), |dist p q - dist (Φ p) (Φ q)| ≤ 2 * (ε₂ / 2 + δ)
x✝ : MetricSpace (X ⊕ Y) := glueMetricApprox (fun x ↦ ↑x) (fun x ↦ Φ x) (ε₂ / 2 + δ) ⋯ this✝³
Fl : X → X ⊕ Y := inl
Fr : Y → X ⊕ Y := inr
Il : Isometry Fl
Ir : Isometry Fr
this✝² : ghDist X Y ≤ hausdorffDist (range Fl) (range Fr)
this✝¹ : hausdorffDist (range Fl) (range Fr) ≤ hausdorffDist (range Fl) (Fl '' s) + hausdorffDist (Fl '' s) (range Fr)
this✝ :
  hausdorffDist (Fl '' s) (range Fr) ≤
    hausdorffDist (Fl '' s) (Fr '' range Φ) + hausdorffDist (Fr '' range Φ) (range Fr)
this : 0 ≤ ε₁
⊢ hausdorffDist univ s ≤ ε₁

Tactic:
refine hausdorffDist_le_of_mem_dist this (fun x _ => hs x) fun x _ =>
      ⟨x, mem_univ _, by simpa only [dist_self]⟩

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

Goal state before:
pp: case refine_3
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 y : ↑(s p)), |dist x y - dist (Φ x) (Φ y)| ≤ ε

Tactic:
intro x y

Goal state after:
case refine_3
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 y : ↑(s p)
⊢ |dist x y - dist (Φ x) (Φ y)| ≤ ε
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 ↦
    (⟨(↑s).base, ⋯⟩, WithTopology.toTopology ⊥ ((ConcreteCategory.hom (F.germ U x hx)) (surjInv ⋯ (↑s).germ)))

Tactic:
refine .prodMk (by fun_prop) ?_
    simp_rw [continuous_iff_continuousAt, ContinuousAt, nhds_discrete, tendsto_pure, nhds_subtype,
      eventually_comap]
    rintro ⟨g, hg⟩
    rcases hF_bij _ hg |>.surjective g.germ with ⟨f, hf⟩
    filter_upwards [g.eventually_nhds hg f hf]
    rintro _ ⟨hgU, hgf⟩ g' rfl
    congr 1
    rw [hgf, ← hf, leftInverse_surjInv (hF_bij _ _), leftInverse_surjInv (hF_bij _ _)]

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
⊢ ∀ (x_1 : ↥U × WithDiscreteTopology (ToType (F.stalk x))),
    ContinuousAt
      (fun x_2 ↦
        ⟨{ base := ↑x_2.1, germ := (ConcreteCategory.hom (F.germ U ↑x_2.1 ⋯)) (surjInv ⋯ x_2.2.ofTopology) }, ⋯⟩)
      x_1

Tactic:
continuousAt_prod_of_discrete_right

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
⊢ ∀ (x_1 : ↥U × WithDiscreteTopology (ToType (F.stalk x))),
    ContinuousAt
      (fun x_2 ↦ ⟨{ base := ↑x_2, germ := (ConcreteCategory.hom (F.germ U ↑x_2 ⋯)) (surjInv ⋯ x_1.2.ofTopology) }, ⋯⟩)
      x_1.1
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)), |dist x y - dist (Φ x) (Φ y)| ≤ ε

Tactic:
intro x y

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)
⊢ |dist x y - dist (Φ x) (Φ y)| ≤ ε
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: X✝ : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X✝ n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X✝ n)
inst✝ : ∀ (n : ℕ), Nonempty (X✝ n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
this : ∀ (n : ℕ), 0 < d n
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
⊢ ∃ x, Tendsto u atTop (𝓝 x)

Tactic:
have ic : ∀ n, Isometry (c n) := fun n x y => by dsimp only [Y, auxGluing]; exact rfl

Goal state after:
X✝ : ℕ → Type
inst✝² : (n : ℕ) → MetricSpace (X✝ n)
inst✝¹ : ∀ (n : ℕ), CompactSpace (X✝ n)
inst✝ : ∀ (n : ℕ), Nonempty (X✝ n)
d : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
this : ∀ (n : ℕ), 0 < d n
u : ℕ → GHSpace
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N
X : ℕ → Type := fun n ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace ⋯ ⋯ = (Y (n + 1)).Space
c : (n : ℕ) → GlueSpace ⋯ ⋯ → (Y (n + 1)).Space := fun n ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
⊢ ∃ x, Tendsto u atTop (𝓝 x)