awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : 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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
n : ℕ
⊢ dist (X3 n) (X3 (n + 1)) ≤ (1 / 2) ^ n

Tactic:
exact le_of_lt (D2 n)

Premises used: le_of_lt
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : 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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
⊢ CauchySeq X3

Tactic:
have : CauchySeq X3 := by
    refine cauchySeq_of_le_geometric (1 / 2) 1 (by norm_num) fun n => ?_
    rw [one_mul]
    exact le_of_lt (D2 n)
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this_1 : ∀ (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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this : CauchySeq X3
rcases : ∀ (L : NonemptyCompacts Z) (hL : Tendsto X3 atTop (𝓝 L)), ∃ (x : GHSpace), Tendsto u atTop (𝓝 x)
⊢ ∃ (x : GHSpace), Tendsto u atTop (𝓝 x)

Tactic:
rcases cauchySeq_tendsto_of_complete this with ⟨L, hL⟩

Premises used: cauchySeq_tendsto_of_complete
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this_1 : ∀ (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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
n : ℕ
rw : Nonempty (↥(X3 n) ≃ᵢ (u n).Rep)
⊢ (NonemptyCompacts.toGHSpace ∘ X3) n = u n

Tactic:
rw [Function.comp_apply, NonemptyCompacts.toGHSpace, ← (u n).toGHSpace_rep,
      toGHSpace_eq_toGHSpace_iff_isometryEquiv]

Premises used: GromovHausdorff.toGHSpace_eq_toGHSpace_iff_isometryEquiv, TopologicalSpace.NonemptyCompacts.toGHSpace, Function.comp_apply, GromovHausdorff.GHSpace.toGHSpace_rep
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this_1 : ∀ (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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
n : ℕ
val : ↥(X3 n) ≃ᵢ (u n).Rep
⊢ Nonempty (↥(X3 n) ≃ᵢ (u n).Rep)

Tactic:
constructor
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this_1 : ∀ (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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
n : ℕ
⊢ ↥(X3 n) ≃ᵢ (u n).Rep

Tactic:
convert! (isom n).isometryEquivOnRange.symm

Premises used: IsometryEquiv.symm, Isometry.isometryEquivOnRange
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this_1 : ∀ (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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
⊢ ∀ (n : ℕ), (NonemptyCompacts.toGHSpace ∘ X3) n = u n

Tactic:
have : ∀ n, (NonemptyCompacts.toGHSpace ∘ X3) n = u n := by
    intro n
    rw [Function.comp_apply, NonemptyCompacts.toGHSpace, ← (u n).toGHSpace_rep,
      toGHSpace_eq_toGHSpace_iff_isometryEquiv]
    constructor
    convert! (isom n).isometryEquivOnRange.symm
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this_1 : ∀ (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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this_2 : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
this : ∀ (n : ℕ), (NonemptyCompacts.toGHSpace ∘ X3) n = u n
h : Tendsto u atTop (𝓝 L.toGHSpace)
⊢ ∃ (x : GHSpace), Tendsto u atTop (𝓝 x)

Tactic:
use L.toGHSpace

Premises used: TopologicalSpace.NonemptyCompacts.toGHSpace
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this_1 : ∀ (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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this_2 : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
this : ∀ (n : ℕ), (NonemptyCompacts.toGHSpace ∘ X3) n = u n
h : Tendsto (NonemptyCompacts.toGHSpace ∘ X3) atTop (𝓝 L.toGHSpace)
⊢ Tendsto u atTop (𝓝 L.toGHSpace)

Tactic:
apply Filter.Tendsto.congr this

Premises used: Filter.Tendsto.congr
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this_1 : ∀ (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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this_2 : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
this : ∀ (n : ℕ), (NonemptyCompacts.toGHSpace ∘ X3) n = u n
⊢ Tendsto (NonemptyCompacts.toGHSpace ∘ X3) atTop (𝓝 L.toGHSpace)

Tactic:
Tendsto.comp ?_ hL

Premises used: Filter.Tendsto.comp
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this_1 : ∀ (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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this_2 : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
this : ∀ (n : ℕ), (NonemptyCompacts.toGHSpace ∘ X3) n = u n
h : Tendsto NonemptyCompacts.toGHSpace (𝓝 L) (𝓝 L.toGHSpace)
⊢ Tendsto (NonemptyCompacts.toGHSpace ∘ X3) atTop (𝓝 L.toGHSpace)

Tactic:
refine Tendsto.comp ?_ hL

Premises used: Filter.Tendsto.comp
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X_1 : ℕ → Type
inst : (n : ℕ) → MetricSpace (X_1 n)
inst_1 : ∀ (n : ℕ), CompactSpace (X_1 n)
inst_2 : ∀ (n : ℕ), Nonempty (X_1 n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this_1 : ∀ (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 : ℕ) → Type := fun (n : ℕ) ↦ (u n).Rep
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E :
  ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))
c : (n : ℕ) →
  (a : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯) → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) :=
  fun (n : ℕ) ↦ cast ⋯
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1)) := fun (n : ℕ) ↦ c n ∘ toGlueL ⋯ ⋯
I : ∀ (n : ℕ), Isometry (f n)
Z0 : Type := InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
Z : Type := UniformSpace.Completion Z0
Φ : (n : ℕ) → (x : (Y n).Space) → InductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I :=
  toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I
coeZ : Z0 → UniformSpace.Completion Z0 := UniformSpace.Completion.coe'
X2 : (n : ℕ) → Set (UniformSpace.Completion Z0) := fun (n : ℕ) ↦ range (coeZ ∘ Φ n ∘ (Y n).embed)
isom : ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)
X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjl (X n) (X n.succ))
X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ optimalGHInjr (X n) (X n.succ))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
X3 : ℕ → NonemptyCompacts Z := fun (n : ℕ) ↦ { carrier := X2 n, isCompact' := ⋯, nonempty' := ⋯ }
this_2 : CauchySeq X3
L : NonemptyCompacts Z
hL : Tendsto X3 atTop (𝓝 L)
this : ∀ (n : ℕ), (NonemptyCompacts.toGHSpace ∘ X3) n = u n
⊢ Tendsto NonemptyCompacts.toGHSpace (𝓝 L) (𝓝 L.toGHSpace)

Tactic:
apply toGHSpace_continuous.tendsto

Premises used: GromovHausdorff.toGHSpace_continuous, Continuous.tendsto
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized
Goal:
X : Type u
Y : Type v
inst : MetricSpace X
inst_1 : MetricSpace Y
⊢ 1 = HMul.hMul (α := ℝ) 2 0 + 1 + HMul.hMul (α := ℝ) 2 0

Tactic:
simp