awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X : ℕ → Type
inst : (n : ℕ) → MetricSpace (X n)
inst_1 : ∀ (n : ℕ), CompactSpace (X n)
inst_2 : ∀ (n : ℕ), Nonempty (X n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
x : ℕ
⊢ 0 < d x

Tactic:
positivity
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X : ℕ → Type
inst : (n : ℕ) → MetricSpace (X n)
inst_1 : ∀ (n : ℕ), CompactSpace (X n)
inst_2 : ∀ (n : ℕ), Nonempty (X n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
⊢ ∀ (n : ℕ), 0 < d n

Tactic:
have : ∀ n : ℕ, 0 < d n := fun _ ↦ by positivity
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X : ℕ → Type
inst : (n : ℕ) → MetricSpace (X n)
inst_1 : ∀ (n : ℕ), CompactSpace (X n)
inst_2 : ∀ (n : ℕ), Nonempty (X n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this : ∀ (n : ℕ), 0 < d n
⊢ CompleteSpace GHSpace

Tactic:
Metric.complete_of_convergent_controlled_sequences d this fun u hu => ?_

Premises used: Metric.complete_of_convergent_controlled_sequences
-- Module: Mathlib.Topology.MetricSpace.GromovHausdorff
Goal:
X : ℕ → Type
inst : (n : ℕ) → MetricSpace (X n)
inst_1 : ∀ (n : ℕ), CompactSpace (X n)
inst_2 : ∀ (n : ℕ), Nonempty (X n)
d : (n : ℕ) → ℝ := fun (n : ℕ) ↦ (1 / 2) ^ n
this : ∀ (n : ℕ), 0 < d n
refine :
  ∀ (u : ℕ → GHSpace) (hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < d N),
    ∃ (x : GHSpace), Tendsto u atTop (𝓝 x)
⊢ CompleteSpace GHSpace

Tactic:
refine Metric.complete_of_convergent_controlled_sequences d this fun u hu => ?_

Premises used: Metric.complete_of_convergent_controlled_sequences
-- 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
n : ℕ
⊢ GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))

Tactic:
dsimp only [Y, auxGluing]

Premises used: GromovHausdorff.auxGluing
-- 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
⊢ ∀ (n : ℕ),
    GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯ = AuxGluingStruct.Space (A := X (n + 1)) (Y (n + 1))

Tactic:
have E :
    ∀ n : ℕ,
      GlueSpace (Y n).isom (isometry_optimalGHInjl (X n) (X (n + 1))) = (Y (n + 1)).Space :=
    fun n => by dsimp only [Y, auxGluing]
-- 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 ⋯
n : ℕ
x y : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯
⊢ edist (c n x) (c n y) = edist x y

Tactic:
rfl

Premises used: setOf, cast, Quotient, SeparationQuotient, Metric.GlueSpace, rfl
-- 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 ⋯
n : ℕ
x y : GlueSpace (Y := OptimalGHCoupling (X n) (X (n + 1))) ⋯ ⋯
⊢ edist (c n x) (c n y) = edist x y

Tactic:
exact rfl

Premises used: rfl
-- 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 ⋯
⊢ ∀ (n : ℕ), Isometry (c n)

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

Tactic:
have I : ∀ n, Isometry (f n) := fun n => (ic n).comp (toGlueL_isometry _ _)
-- 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)
n : ℕ
⊢ Isometry (coeZ ∘ Φ n ∘ (Y n).embed)

Tactic:
UniformSpace.Completion.coe_isometry.comp ?_

Premises used: Isometry.comp, UniformSpace.Completion.coe_isometry
-- 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)
n : ℕ
refine : Isometry (Φ n ∘ (Y n).embed)
⊢ Isometry (coeZ ∘ Φ n ∘ (Y n).embed)

Tactic:
refine UniformSpace.Completion.coe_isometry.comp ?_

Premises used: Isometry.comp, UniformSpace.Completion.coe_isometry
-- 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)
n : ℕ
⊢ Isometry (Φ n ∘ (Y n).embed)

Tactic:
(toInductiveLimit_isometry _ _).comp (Y n).isom

Premises used: Isometry.comp, GromovHausdorff.AuxGluingStruct.isom, Metric.toInductiveLimit_isometry
-- 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)
n : ℕ
⊢ Isometry (Φ n ∘ (Y n).embed)

Tactic:
exact (toInductiveLimit_isometry _ _).comp (Y n).isom

Premises used: Isometry.comp, Metric.toInductiveLimit_isometry
-- 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)
⊢ ∀ (n : ℕ), Isometry (coeZ ∘ Φ n ∘ (Y n).embed)

Tactic:
have isom : ∀ n, Isometry (coeZ ∘ Φ n ∘ (Y n).embed) := by
    intro n
    refine UniformSpace.Completion.coe_isometry.comp ?_
    exact (toInductiveLimit_isometry _ _).comp (Y n).isom
-- 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)
n : ℕ
change : X2 n = range (coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯ ∘ optimalGHInjl (X n) (X n.succ))
⊢ 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))

Premises used: Function.comp, GromovHausdorff.AuxGluingStruct.isom, GromovHausdorff.optimalGHInjl, Set.range, Nat.succ, GromovHausdorff.isometry_optimalGHInjl, Metric.toGlueR
-- 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)
n : ℕ
simp :
  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))
⊢ X2 n = range (coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯ ∘ optimalGHInjl (X n) (X n.succ))

Tactic:
simp only [X2, Φ]
-- 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)
n : ℕ
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 ⋯ ⋯ ∘ optimalGHInjl (X n) (X n.succ))
⊢ 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))

Tactic:
rw [← toInductiveLimit_commute I]

Premises used: Metric.toInductiveLimit_commute
-- 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)
n : ℕ
⊢ 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 ⋯ ⋯ ∘ optimalGHInjl (X n) (X n.succ))

Tactic:
simp only [f, ← toGlue_commute, Function.comp_assoc]

Premises used: Metric.toGlue_commute, Function.comp_assoc
-- 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)
⊢ ∀ (n : ℕ), X2 n = range ((coeZ ∘ Φ n.succ ∘ c n ∘ toGlueR ⋯ ⋯) ∘ 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]