awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
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)
n : ℕ
⊢ range (coeZ ∘ toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I n ∘ (Y n).embed) =
    range
      (coeZ ∘
        toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I n.succ ∘ c n ∘ toGlueR ⋯ ⋯ ∘ optimalGHInjl (X n) (X n.succ))
have rw : range (coeZ ∘ (toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I n.succ ∘ f n) ∘ (Y n).embed) =
  range
    (coeZ ∘
      toInductiveLimit (X := fun (n : ℕ) ↦ (Y n).Space) I n.succ ∘ c n ∘ toGlueR sorry sorry ∘ optimalGHInjl (X n) (X n.succ))
Tactic: rw [← toInductiveLimit_commute I]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (hypothesis)
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)
⊢ ∃ (x : GHSpace), Tendsto u atTop (𝓝 x)
have X2n : ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR sorry sorry) ∘ optimalGHInjl (X n) (X n.succ))
Tactic: have X2n : ∀ n, 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)) := by
    intro n
    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))
    simp only [X2, Φ]
    rw [← toInductiveLimit_commute I]
    simp only [f, ← toGlue_commute, Function.comp_assoc]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (hypothesis)
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))
⊢ ∃ (x : GHSpace), Tendsto u atTop (𝓝 x)
have X2nsucc : ∀ (n : ℕ), X2 n.succ = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR sorry sorry) ∘ optimalGHInjr (X n) (X n.succ))
Tactic: have X2nsucc : ∀ n, X2 n.succ =
      range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR (Y n).isom
        (isometry_optimalGHInjl (X n) (X n.succ))) ∘ optimalGHInjr (X n) (X n.succ)) := by
    intro n
    rfl
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
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))
n : ℕ
⊢ hausdorffDist (X2 n) (X2 n.succ) < d n
have rw : dist (u n) (u n.succ) < d n
Tactic: rw [X2n n, X2nsucc n, range_comp, range_comp, hausdorffDist_image,
      hausdorffDist_optimal, ← dist_ghDist]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
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))
n : ℕ
rw : dist (u n) (u n.succ) < d n
⊢ hausdorffDist (X2 n) (X2 n.succ) < d n
have rw₁ : Isometry (coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR sorry sorry)
Tactic: rw [X2n n, X2nsucc n, range_comp, range_comp, hausdorffDist_image,
      hausdorffDist_optimal, ← dist_ghDist]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
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))
n : ℕ
⊢ Isometry (coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯)
have apply : Isometry (Φ n.succ ∘ c n ∘ toGlueR sorry sorry)
Tactic: apply UniformSpace.Completion.coe_isometry.comp _
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (hypothesis)
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))
⊢ ∃ (x : GHSpace), Tendsto u atTop (𝓝 x)
have D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 n.succ) < d n
Tactic: have D2 : ∀ n, hausdorffDist (X2 n) (X2 n.succ) < d n := fun n ↦ by
    rw [X2n n, X2nsucc n, range_comp, range_comp, hausdorffDist_image,
      hausdorffDist_optimal, ← dist_ghDist]
    · exact hu n n n.succ (le_refl n) (le_succ n)
    · apply UniformSpace.Completion.coe_isometry.comp _
      exact (toInductiveLimit_isometry _ _).comp ((ic n).comp (toGlueR_isometry _ _))
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
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
have refine : ∀ (n : ℕ), dist (X3 n) (X3 (n + 1)) ≤ 1 * (1 / 2) ^ n
Tactic: refine cauchySeq_of_le_geometric (1 / 2) 1 (by norm_num) fun n => ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
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 * (1 / 2) ^ n
have rw : dist (X3 n) (X3 (n + 1)) ≤ (1 / 2) ^ n
Tactic: rw [one_mul]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (hypothesis)
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' := ⋯ }
⊢ ∃ (x : GHSpace), Tendsto u atTop (𝓝 x)
have this₁ : 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)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
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
⊢ ∃ (x : GHSpace), Tendsto u atTop (𝓝 x)
have rcases : ∀ (L : NonemptyCompacts Z) (hL : Tendsto X3 atTop (𝓝 L)), ∃ (x : GHSpace), Tendsto u atTop (𝓝 x)
Tactic: rcases cauchySeq_tendsto_of_complete this with ⟨L, hL⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.GromovHausdorff (goal)
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
have rw : Nonempty (↥(X3 n) ≃ᵢ (u n).Rep)
Tactic: rw [Function.comp_apply, NonemptyCompacts.toGHSpace, ← (u n).toGHSpace_rep,
      toGHSpace_eq_toGHSpace_iff_isometryEquiv]