mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.EMetricSpace.PairReduction

Goal state before:
pp: case pos
T : Type u_1
inst✝¹ : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst✝ : DecidableEq T
hJ : J.Nonempty
ha : 1 < a
i : ℕ
ih :
  ↑(#(pairSetSeq J a c i)) ≤
    (if (logSizeBallSeq J hJ a c i).finset.Nonempty then 1 else 0) * a ^ (logSizeBallSeq J hJ a c i).radius
h : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty
⊢ ↑(#(pairSetSeq J a c (i + 1))) ≤
    (if (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty then 1 else 0) * a ^ (logSizeBallSeq J hJ a c (i + 1)).radius

Tactic:
simpa [pairSetSeq, logSizeBallStruct.ball, h, hJ]
        using! card_le_logSizeRadius_le_pow_logSizeRadius ha

Goal state after:
Module: Mathlib.Topology.Homotopy.HomotopyGroup

Goal state before:
pp: N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
f : ↑(Ω^ N X x)
⊢ ⟦symmAt i f⟧ = ⟦symmAt j f⟧

Tactic:
← fromLoop_symm_toLoop

Goal state after:
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
f : ↑(Ω^ N X x)
⊢ ⟦fromLoop i (Path.symm (toLoop i f))⟧ = ⟦fromLoop j (Path.symm (toLoop j f))⟧
Module: Mathlib.Topology.Homotopy.HomotopyGroup

Goal state before:
pp: N : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N
inst✝ : Nonempty N
i : N
p : ↑(Ω^ N X x)
⊢ (⟦p⟧)⁻¹ = ⟦symmAt (Classical.arbitrary N) p⟧

Tactic:
← fromLoop_symm_toLoop

Goal state after:
N : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
x : X
inst✝¹ : DecidableEq N
inst✝ : Nonempty N
i : N
p : ↑(Ω^ N X x)
⊢ (⟦p⟧)⁻¹ = ⟦fromLoop (Classical.arbitrary N) (Path.symm (toLoop (Classical.arbitrary N) p))⟧
Module: Mathlib.Topology.MetricSpace.Kuratowski

Goal state before:
pp: case h
α : Type u
n : ℕ
inst✝ : MetricSpace α
x : ℕ → α
a : α
⊢ dist a (x 0) ∈ upperBounds (range fun i ↦ ‖dist a (x i) - dist (x 0) (x i)‖)

Tactic:
rintro - ⟨n, rfl⟩

Goal state after:
case h
α : Type u
n✝ : ℕ
inst✝ : MetricSpace α
x : ℕ → α
a : α
n : ℕ
⊢ (fun i ↦ ‖dist a (x i) - dist (x 0) (x i)‖) n ≤ dist a (x 0)
Module: Mathlib.Topology.MetricSpace.Kuratowski

Goal state before:
pp: case inr
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
ι : Type u_1
f : α → ↥(lp (fun x ↦ ℝ) ∞)
K : ℝ≥0
hfl : ∀ (i : ι), LipschitzOnWith K (fun a ↦ ↑(f a) i) s
g : ι → α → ℝ
hgl : ∀ (i : ι), LipschitzWith K (g i)
hgeq : ∀ (i : ι), EqOn (fun x ↦ ↑(f x) i) (g i) s
a₀ : α
ha₀_in_s : a₀ ∈ s
hf_extb : ∀ (a : α), Memℓp (swap g a) ∞
⊢ ∃ g, LipschitzWith K g ∧ EqOn f g s

Tactic:
let f_ext' : α → ℓ^∞(ι, ℝ) := fun i ↦ ⟨swap g i, hf_extb i⟩

Goal state after:
case inr
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
ι : Type u_1
f : α → ↥(lp (fun x ↦ ℝ) ∞)
K : ℝ≥0
hfl : ∀ (i : ι), LipschitzOnWith K (fun a ↦ ↑(f a) i) s
g : ι → α → ℝ
hgl : ∀ (i : ι), LipschitzWith K (g i)
hgeq : ∀ (i : ι), EqOn (fun x ↦ ↑(f x) i) (g i) s
a₀ : α
ha₀_in_s : a₀ ∈ s
hf_extb : ∀ (a : α), Memℓp (swap g a) ∞
f_ext' : α → ↥(lp (fun x ↦ ℝ) ∞) := fun i ↦ ⟨swap g i, ⋯⟩
⊢ ∃ g, LipschitzWith K g ∧ EqOn f g s
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : EMetricSpace α
inst✝ : EMetricSpace β
s✝ : Set α
s t : Compacts α
h : edist s t = 0
⊢ ↑s = ↑t

Tactic:
have : closure (s : Set α) = closure t := hausdorffEDist_zero_iff_closure_eq_closure.1 h

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : EMetricSpace α
inst✝ : EMetricSpace β
s✝ : Set α
s t : Compacts α
h : edist s t = 0
this :
  closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] ↑s =
    closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] ↑t
⊢ ↑s = ↑t
Module: Mathlib.Topology.MetricSpace.Closeds

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : EMetricSpace α
inst✝ : EMetricSpace β
s✝ : Set α
s t : NonemptyCompacts α
h : edist s t = 0
⊢ ↑s = ↑t

Tactic:
have : closure (s : Set α) = closure t := hausdorffEDist_zero_iff_closure_eq_closure.1 h

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : EMetricSpace α
inst✝ : EMetricSpace β
s✝ : Set α
s t : NonemptyCompacts α
h : edist s t = 0
this :
  closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] ↑s =
    closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] ↑t
⊢ ↑s = ↑t
Module: Mathlib.Topology.MetricSpace.Infsep

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Decidable s.Nontrivial
hs : s.Nontrivial
d : ℝ
h : ∃ a b, (a, b) ∈ s.offDiag ∧ uncurry dist (a, b) = d
⊢ 0 ≤ d

Tactic:
uncurry_apply_pair

Goal state after:
α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Decidable s.Nontrivial
hs : s.Nontrivial
d : ℝ
h : ∃ a b, (a, b) ∈ s.offDiag ∧ dist a b = d
⊢ 0 ≤ d
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: case refine_1
X : Type u
inst✝² : MetricSpace X
inst✝¹ : CompactSpace X
inst✝ : Nonempty X
p : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)
h : IsometryRel.setoid p (NonemptyCompacts.kuratowskiEmbedding X)
⊢ ∃ Ψ, Isometry Ψ ∧ range Ψ = ↑p

Tactic:
rcases Setoid.symm h with ⟨e⟩
    have f := (kuratowskiEmbedding.isometry X).isometryEquivOnRange.trans e
    use fun x => f x, isometry_subtype_coe.comp f.isometry
    rw [range_comp', f.range_eq_univ, Set.image_univ, Subtype.range_coe]

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

Goal state before:
pp: α : Type u
ι : Type v
inst✝¹ : MetricSpace α
inst✝ : ProperSpace α
c : ι → α
s : Set α
r : ι → ℝ
hs : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
uf : ∀ x ∈ s, {i | x ∈ ball (c i) (r i)}.Finite
us : s ⊆ ⋃ i, ball (c i) (r i)
⊢ ∃ r', s ⊆ ⋃ i, ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i

Tactic:
rcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with
    ⟨v, hsv, hvc, hcv⟩
  have := fun i => exists_lt_subset_ball (hvc i) (hcv i)
  choose r' hlt hsub using this
  exact ⟨r', hsv.trans <| iUnion_mono <| hsub, hlt⟩

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

Goal state before:
pp: case pos
α : Type u_1
inst✝ : PseudoMetricSpace α
s : Finset α
H : (↑s).Nontrivial ↔ s.offDiag.Nonempty
hs : s.offDiag.Nonempty
⊢ (↑s).infsep = s.offDiag.inf' hs (uncurry dist)

Tactic:
simp_rw [(H.mpr hs).infsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]

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

Goal state before:
pp: case intro
α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Finite ↑s
hs : s.Nontrivial
val✝ : Fintype ↑s
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y

Tactic:
hs.infsep_of_fintype

Goal state after:
case intro
α : Type u_1
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : Finite ↑s
hs : s.Nontrivial
val✝ : Fintype ↑s
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' ⋯ (uncurry dist) = dist x y
Module: Mathlib.Topology.Order.UpperLowerSetTopology

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : Preorder α
inst✝ : TopologicalSpace α
⊢ Topology.IsLowerSet α → Topology.IsUpperSet αᵒᵈ

Tactic:
· apply OrderDual.instIsUpperSet

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

Goal state before:
pp: α : Type u_1
β : Type u_2
D : Set (Set α)
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
inst✝ : IsScott α D
hf : Continuous[inst✝⁴, inst✝²] f
x✝ b : α
hab : x✝ ≤ b
⊢ f x✝ ≤ f b

Tactic:
by_contra h

Goal state after:
α : Type u_1
β : Type u_2
D : Set (Set α)
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
inst✝ : IsScott α D
hf : Continuous[inst✝⁴, inst✝²] f
x✝ b : α
hab : x✝ ≤ b
h : ¬f x✝ ≤ f b
⊢ False
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
D : Set (Set α)
inst✝ : IsScott α D
hD : ∀ (a b : α), a ≤ b → {a, b} ∈ D
hf : Continuous[inst✝⁴, inst✝²] f
t : Set α
h₀ : t ∈ D
d₁ : t.Nonempty
d₂ : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) t
a : α
d₃ : IsLUB t a
b : β
hb : b ∈ upperBounds (f '' t)
⊢ f a ≤ b

Tactic:
by_contra h

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : TopologicalSpace α
inst✝³ : Preorder β
inst✝² : TopologicalSpace β
inst✝¹ : IsScott β univ
f : α → β
D : Set (Set α)
inst✝ : IsScott α D
hD : ∀ (a b : α), a ≤ b → {a, b} ∈ D
hf : Continuous[inst✝⁴, inst✝²] f
t : Set α
h₀ : t ∈ D
d₁ : t.Nonempty
d₂ : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) t
a : α
d₃ : IsLUB t a
b : β
hb : b ∈ upperBounds (f '' t)
h : ¬f a ≤ b
⊢ False
Module: Mathlib.Topology.Order.LawsonTopology

Goal state before:
pp: case e'_2
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLawson α
lawsonBasis_image2 :
  lawsonBasis α =
    image2 (fun x x_1 ↦ ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))
      {U | IsOpen[scott α univ] U}
⊢ inst✝¹ =
    induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓
      induced (⇑WithScott.toScott) WithScott.instTopologicalSpace

Tactic:
@topology_eq_lawson α _ _ _,

Goal state after:
case e'_2
α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLawson α
lawsonBasis_image2 :
  lawsonBasis α =
    image2 (fun x x_1 ↦ ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))
      {U | IsOpen[scott α univ] U}
⊢ lawson α =
    induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓
      induced (⇑WithScott.toScott) WithScott.instTopologicalSpace
Module: Mathlib.Topology.Order.LawsonTopology

Goal state before:
pp: α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : TopologicalSpace α
inst✝ : IsLawson α
a : α
⊢ IsClosed[inst✝¹] {a}

Tactic:
simp +instances only [IsLawson.topology_eq_lawson]
    rw [← (Set.OrdConnected.upperClosure_inter_lowerClosure ordConnected_singleton),
      ← WithLawson.isClosed_preimage_ofLawson]
    apply IsClosed.inter
      (lawsonClosed_of_lowerClosed _ (IsLower.isClosed_upperClosure (finite_singleton a)))
    rw [lowerClosure_singleton, LowerSet.coe_Iic, ← WithLawson.isClosed_preimage_ofLawson]
    exact lawsonClosed_of_scottClosed _ isClosed_Iic

Goal state after:
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)
hΦ : IsCompact (range Φ)
hΨ : IsCompact (range Ψ)
⊢ range Φ = range Ψ

Tactic:
apply (IsClosed.hausdorffDist_zero_iff_eq _ _ _).1 DΦΨ.symm

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)
hΦ : IsCompact (range Φ)
hΨ : IsCompact (range Ψ)
⊢ IsClosed (range Φ)

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)
hΦ : IsCompact (range Φ)
hΨ : IsCompact (range Ψ)
⊢ IsClosed (range Ψ)

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)
hΦ : IsCompact (range Φ)
hΨ : IsCompact (range Ψ)
⊢ hausdorffEDist (range Φ) (range Ψ) ≠ ∞
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝⁸ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝² : TopologicalSpace F
inst✝¹ : AddCommGroup F
inst✝ : Module ℝ F
cY : Convex ℝ Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
a : E
b b' : β
y y' : ↑Y
ha : ∀ x ∈ X, max (f a ↑y) (f a ↑y') ≤ max (f x ↑y) (f x ↑y')
hb : ∀ y ∈ Y, ∃ x ∈ X, f x y ≤ b
hb' : b' < max (f a ↑y) (f a ↑y')
hbb' : b < b'
J : Set ↑(segment ℝ ↑y ↑y') := setOfPred_sublevelLeft_subset X Y f b b' y y'
z : ↑(segment ℝ ↑y ↑y')
hz : ClusterPt z (𝓟 J)
x : ↑X
hx : x ∈ sublevelLeft X f b ↑z
hfx : ∀ x ∈ X, ∀ x_1 ∈ Y, ∀ (y : β), f x x_1 < y → ∀ᶠ (x' : F) in nhdsWithin x_1 Y, f x x' < y
⊢ ∃ z' ∈ setOfPred_sublevelLeft_subset X Y f b b' y y', x ∈ sublevelLeft X f b' ↑z'

Tactic:
specialize hfx x.val x.prop z (cY.segment_subset y.prop y'.prop z.prop) b' (hx.trans_lt hbb')

Goal state after:
E : Type u_1
F : Type u_2
β : Type u_3
inst✝⁸ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝⁷ : TopologicalSpace E
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul ℝ E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝² : TopologicalSpace F
inst✝¹ : AddCommGroup F
inst✝ : Module ℝ F
cY : Convex ℝ Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
a : E
b b' : β
y y' : ↑Y
ha : ∀ x ∈ X, max (f a ↑y) (f a ↑y') ≤ max (f x ↑y) (f x ↑y')
hb : ∀ y ∈ Y, ∃ x ∈ X, f x y ≤ b
hb' : b' < max (f a ↑y) (f a ↑y')
hbb' : b < b'
J : Set ↑(segment ℝ ↑y ↑y') := setOfPred_sublevelLeft_subset X Y f b b' y y'
z : ↑(segment ℝ ↑y ↑y')
hz : ClusterPt z (𝓟 J)
x : ↑X
hx : x ∈ sublevelLeft X f b ↑z
hfx : ∀ᶠ (x' : F) in nhdsWithin (↑z) Y, f (↑x) x' < b'
⊢ ∃ z' ∈ setOfPred_sublevelLeft_subset X Y f b b' y y', x ∈ sublevelLeft X f b' ↑z'
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 ⋯
n : ℕ
x y : GlueSpace ⋯ ⋯
⊢ edist (c n x) (c n y) = edist x y

Tactic:
dsimp only [Y, auxGluing]; exact rfl

Goal state after:
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 ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (Y n).Space → InductiveLimit I := toInductiveLimit I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : ℕ → Set (UniformSpace.Completion Z0) := fun n ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
n : ℕ
⊢ X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))

Tactic:
change X2 n = range (coeZ ∘ Φ n.succ ∘ c n ∘
      toGlueR (Y n).isom (isometry_optimalGHInjl (X n) (X n.succ)) ∘
      optimalGHInjl (X n) (X n.succ))

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)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space := fun n ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (Y n).Space → InductiveLimit I := toInductiveLimit I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : ℕ → Set (UniformSpace.Completion Z0) := fun n ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
n : ℕ
⊢ X2 n = range (coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯ ∘ optimalGHInjl (X n) (X n.succ))
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: case neg.refine_1
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : QuasiSober X
inst✝¹ : PrespectralSpace X
inst✝ : QuasiSeparatedSpace X
𝔅 : Set (Set X) := constructibleTopologySubbasis X
𝒮 : Set (Set (Set X)) := {P | P ⊆ 𝔅 ∧ (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) ∧ ⋂₀ P = ∅}
s : Set (Set X)
hs : s ∈ 𝒮
B : Set (Set X)
hsB : s ⊆ B
hB : Maximal (fun x ↦ x ∈ 𝒮) B
B' : Set (Set X) := {s | s ∈ B ∧ IsClosed[inst✝⁴] s}
Z : Set X := ⋂₀ B'
hZ_closed : IsClosed[inst✝⁴] Z
hZ_nonempty : Z.Nonempty
hZ_irred : ¬IsPreirreducible Z
Y₁ Y₂ : Set X
hY₁_cl : IsClosed[inst✝⁴] Y₁
hY₂_cl : IsClosed[inst✝⁴] Y₂
hc₁ : IsCompact Y₁ᶜ
hc₂ : IsCompact Y₂ᶜ
hsub : Z ⊆ Y₁ ∪ Y₂
x₁ : X
hx₁ : x₁ ∈ Z ∩ Y₁ᶜ
x₂ : X
hx₂ : x₂ ∈ Z ∩ Y₂ᶜ
hY₁ : Y₁ ∈ 𝔅
hY₂ : Y₂ ∈ 𝔅
hY₁_inter_B : Y₁ ∩ ⋂₀ B = ∅
hY₂_inter_B : Y₂ ∩ ⋂₀ B = ∅
A₁ : Set (Set X)
hA₁ : A₁ ⊆ insert Y₁ B
hA₁' : A₁.Finite
hA₁'' : ⋂₀ A₁ = ∅
A₂ : Set (Set X)
hA₂ : A₂ ⊆ insert Y₂ B
hA₂' : A₂.Finite
hA₂'' : ⋂₀ A₂ = ∅
this : ¬(Z ∩ ⋂₀ (A₁ \ {Y₁} ∪ A₂ \ {Y₂})).Nonempty
⊢ IsCompact (⋂₀ (A₁ \ {Y₁} ∪ A₂ \ {Y₂}))

Tactic:
apply isCompact_sInter_of_subset_constructibleTopologySubbasis _ (hA₁'.sdiff.union hA₂'.sdiff)
      grind [hB.prop, Set.union_subset_iff]

Goal state after:
Module: Mathlib.Topology.UniformSpace.Ultra.Constructions

Goal state before:
pp: X : Type u_1
u u' : UniformSpace X
h : IsUltraUniformity X
h' : IsUltraUniformity X
⊢ IsUltraUniformity X

Tactic:
let := u ⊓ u'

Goal state after:
X : Type u_1
u u' : UniformSpace X
h : IsUltraUniformity X
h' : IsUltraUniformity X
this : UniformSpace X := u ⊓ u'
⊢ IsUltraUniformity X
Module: Mathlib.Topology.UniformSpace.Ultra.Completion

Goal state before:
pp: X : Type u_1
inst✝ : UniformSpace X
⊢ IsUltraUniformity (UniformSpace.Completion X) ↔ IsUltraUniformity X

Tactic:
iff_comm,

Goal state after:
X : Type u_1
inst✝ : UniformSpace X
⊢ IsUltraUniformity X ↔ IsUltraUniformity (UniformSpace.Completion X)