/-- The Gromov-Hausdorff space is complete. -/
instance : CompleteSpace GHSpace := by
  set d := fun n : ℕ ↦ ((1 : ℝ) / 2) ^ n
  have : ∀ n : ℕ, 0 < d n := fun _ ↦ by positivity
  -- start from a sequence of nonempty compact metric spaces within distance `1/2^n` of each other
  refine' Metric.complete_of_convergent_controlled_sequences d this fun u hu => _
  -- `X n` is a representative of `u n`
  let X n := (u n).Rep
  -- glue them together successively in an optimal way, getting a sequence of metric spaces `Y n`
  let Y := auxGluing X
  -- this equality is true by definition but Lean unfolds some defs in the wrong order
  have E :
    ∀ n : ℕ,
      GlueSpace (Y n).isom (isometry_optimalGHInjl (X n) (X (n + 1))) = (Y (n + 1)).Space :=
    fun n => by dsimp only [auxGluing]; rfl
  let c n := cast (E n)
  have ic : ∀ n, Isometry (c n) := fun n x y => by dsimp only [auxGluing]; exact rfl
  -- there is a canonical embedding of `Y n` in `Y (n+1)`, by construction
  let f : ∀ n, (Y n).Space → (Y (n + 1)).Space := fun n =>
    c n ∘ toGlueL (Y n).isom (isometry_optimalGHInjl (X n) (X n.succ))
  have I : ∀ n, Isometry (f n) := fun n => (ic n).comp (toGlueL_isometry _ _)
  -- consider the inductive limit `Z0` of the `Y n`, and then its completion `Z`
  let Z0 := Metric.InductiveLimit I
  let Z := UniformSpace.Completion Z0
  let Φ := toInductiveLimit I
  let coeZ := ((↑) : Z0 → Z)
  -- let `X2 n` be the image of `X n` in the space `Z`
  let X2 n := range (coeZ ∘ Φ n ∘ (Y n).embed)
  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
  -- The Hausdorff distance of `X2 n` and `X2 (n+1)` is by construction the distance between
  -- `u n` and `u (n+1)`, therefore bounded by `1/2^n`
  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]
    rw [← toGlue_commute]
    rfl
  -- simp_rw [range_comp] at X2n
  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
  -- simp_rw [range_comp] at X2nsucc
  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 _ _))
  -- consider `X2 n` as a member `X3 n` of the type of nonempty compact subsets of `Z`, which
  -- is a metric space
  let X3 : ℕ → NonemptyCompacts Z := fun n =>
    ⟨⟨X2 n, isCompact_range (isom n).continuous⟩, range_nonempty _⟩
  -- `X3 n` is a Cauchy sequence by construction, as the successive distances are
  -- bounded by `(1/2)^n`
  have : CauchySeq X3 := by
    refine' cauchySeq_of_le_geometric (1 / 2) 1 (by norm_num) fun n => _
    rw [one_mul]
    

Proof state:
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 => GHSpace.Rep (u n)
Y : (n : ℕ) → AuxGluingStruct (X n) := auxGluing X
E : ∀ (n : ℕ), GlueSpace (_ : Isometry (Y n).embed) (_ : Isometry (optimalGHInjl (X n) (X (n + 1)))) = (Y (n + 1)).Space
c : (n : ℕ) →
  GlueSpace (_ : Isometry (Y n).embed) (_ : Isometry (optimalGHInjl (X n) (X (n + 1)))) → (Y (n + 1)).Space :=
  fun n =>
    cast (_ : GlueSpace (_ : Isometry (Y n).embed) (_ : Isometry (optimalGHInjl (X n) (X (n + 1)))) = (Y (n + 1)).Space)
ic : ∀ (n : ℕ), Isometry (c n)
f : (n : ℕ) → (Y n).Space → (Y (n + 1)).Space :=
  fun n => c n ∘ toGlueL (_ : Isometry (Y n).embed) (_ : Isometry (optimalGHInjl (X n) (X (Nat.succ n))))
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 := ↑Z0
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)
X2n :
  ∀ (n : ℕ),
    X2 n =
      range
        ((coeZ ∘
            Φ (Nat.succ n) ∘
              c n ∘ toGlueR (_ : Isometry (Y n).embed) (_ : Isometry (optimalGHInjl (X n) (X (Nat.succ n))))) ∘
          optimalGHInjl (X n) (X (Nat.succ n)))
X2nsucc :
  ∀ (n : ℕ),
    X2 (Nat.succ n) =
      range
        ((coeZ ∘
            Φ (Nat.succ n) ∘
              c n ∘ toGlueR (_ : Isometry (Y n).embed) (_ : Isometry (optimalGHInjl (X n) (X (Nat.succ n))))) ∘
          optimalGHInjr (X n) (X (Nat.succ n)))
D2 : ∀ (n : ℕ), hausdorffDist (X2 n) (X2 (Nat.succ n)) < d n
X3 : ℕ → NonemptyCompacts Z :=
  fun n =>
    { toCompacts := { carrier := X2 n, isCompact' := (_ : IsCompact (range (coeZ ∘ Φ n ∘ (Y n).embed))) },
      nonempty' := (_ : Set.Nonempty (range fun y => (coeZ ∘ Φ n ∘ (Y n).embed) y)) }
n : ℕ
⊢ dist (X3 n) (X3 (n + 1)) ≤ (1 / 2) ^ n

Next tactic:
exact le_of_lt (D2 n)