-- Mathlib/Topology/MetricSpace/Perfect.lean
theorem Perfect.exists_nat_bool_injection
    (hC : Perfect C) (hnonempty : C.Nonempty) [CompleteSpace α] :
    ∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Injective f

Proof state:
case hfh.intro
α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥0∞} → 0 < ε → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 hC hnonempty hε) ∧
      (C0 hC hnonempty hε).Nonempty ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 hC hnonempty hε) ∧
      (C1 hC hnonempty hε).Nonempty ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)
DP : List Bool → P :=
  fun l =>
    List.rec ⟨C, ⋯⟩
      (fun a l ih => Bool.casesOn (motive := fun t => a = t → P) a (fun h => ⟨C0 ⋯ ⋯ ⋯, ⋯⟩) (fun h => ⟨C1 ⋯ ⋯ ⋯, ⋯⟩) ⋯)
      l
D : List Bool → Set α := fun l => ↑(DP l)
hanti : ClosureAntitone D
x : ℕ → Bool
n : ℕ
hm : n.succ ≠ 0
⊢ EMetric.diam (D (x n :: PiNat.res x n)) ≤ u (n + 1)

Tactic:
cases x n

Result:
case hfh.intro.false
α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥0∞} → 0 < ε → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 hC hnonempty hε) ∧
      (C0 hC hnonempty hε).Nonempty ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 hC hnonempty hε) ∧
      (C1 hC hnonempty hε).Nonempty ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)
DP : List Bool → P :=
  fun l =>
    List.rec ⟨C, ⋯⟩
      (fun a l ih => Bool.casesOn (motive := fun t => a = t → P) a (fun h => ⟨C0 ⋯ ⋯ ⋯, ⋯⟩) (fun h => ⟨C1 ⋯ ⋯ ⋯, ⋯⟩) ⋯)
      l
D : List Bool → Set α := fun l => ↑(DP l)
hanti : ClosureAntitone D
x : ℕ → Bool
n : ℕ
hm : n.succ ≠ 0
⊢ EMetric.diam (D (false :: PiNat.res x n)) ≤ u (n + 1)

case hfh.intro.true
α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥0∞} → 0 < ε → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 hC hnonempty hε) ∧
      (C0 hC hnonempty hε).Nonempty ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 hC hnonempty hε) ∧
      (C1 hC hnonempty hε).Nonempty ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)
DP : List Bool → P :=
  fun l =>
    List.rec ⟨C, ⋯⟩
      (fun a l ih => Bool.casesOn (motive := fun t => a = t → P) a (fun h => ⟨C0 ⋯ ⋯ ⋯, ⋯⟩) (fun h => ⟨C1 ⋯ ⋯ ⋯, ⋯⟩) ⋯)
      l
D : List Bool → Set α := fun l => ↑(DP l)
hanti : ClosureAntitone D
x : ℕ → Bool
n : ℕ
hm : n.succ ≠ 0
⊢ EMetric.diam (D (true :: PiNat.res x n)) ≤ u (n + 1)

Proof state:
α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥0∞} → 0 < ε → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 hC hnonempty hε) ∧
      (C0 hC hnonempty hε).Nonempty ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 hC hnonempty hε) ∧
      (C1 hC hnonempty hε).Nonempty ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)
DP : List Bool → P :=
  fun l =>
    List.rec ⟨C, ⋯⟩
      (fun a l ih => Bool.casesOn (motive := fun t => a = t → P) a (fun h => ⟨C0 ⋯ ⋯ ⋯, ⋯⟩) (fun h => ⟨C1 ⋯ ⋯ ⋯, ⋯⟩) ⋯)
      l
D : List Bool → Set α := fun l => ↑(DP l)
hanti : ClosureAntitone D
hdiam : VanishingDiam D
hdisj' : CantorScheme.Disjoint D
x : ℕ → Bool
⊢ x ∈ (inducedMap D).fst

Tactic:
rw [hanti.map_of_vanishingDiam hdiam fun l => (DP l).property.2]

Result:
α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥0∞} → 0 < ε → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 hC hnonempty hε) ∧
      (C0 hC hnonempty hε).Nonempty ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 hC hnonempty hε) ∧
      (C1 hC hnonempty hε).Nonempty ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)
DP : List Bool → P :=
  fun l =>
    List.rec ⟨C, ⋯⟩
      (fun a l ih => Bool.casesOn (motive := fun t => a = t → P) a (fun h => ⟨C0 ⋯ ⋯ ⋯, ⋯⟩) (fun h => ⟨C1 ⋯ ⋯ ⋯, ⋯⟩) ⋯)
      l
D : List Bool → Set α := fun l => ↑(DP l)
hanti : ClosureAntitone D
hdiam : VanishingDiam D
hdisj' : CantorScheme.Disjoint D
x : ℕ → Bool
⊢ x ∈ univ