-- Mathlib/Computability/PostTuringMachine.lean
theorem supportsStmt_read {S : Finset (Λ' Γ Λ σ)} :
    ∀ {f : Γ → Stmt Bool (Λ' Γ Λ σ) σ}, (∀ a, SupportsStmt S (f a)) →
      SupportsStmt S (read dec f) :=
  suffices
    ∀ (i) (f : List.Vector Bool i → Stmt Bool (Λ' Γ Λ σ) σ),
      (∀ v, SupportsStmt S (f v)) → SupportsStmt S (readAux i f)
    from fun hf ↦ this n _ (by intro; simp only [supportsStmt_move, hf])
  fun i f hf ↦ by
  induction' i with i IH; · exact hf _
  constructor <;> apply IH <;> intro <;> apply hf

Proof state:
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
dec : List.Vector Bool n → Γ
S : Finset (Λ' Γ Λ σ)
f✝ : Γ → Stmt Bool (Λ' Γ Λ σ) σ
this :
  ∀ (i : ℕ) (f : List.Vector Bool i → Stmt Bool (Λ' Γ Λ σ) σ),
    (∀ (v : List.Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f)
hf : ∀ (a : Γ), SupportsStmt S (f✝ a)
⊢ ∀ (v : List.Vector Bool n), SupportsStmt S (move n Dir.left (f✝ (dec v)))

Tactic:
intro

Result:
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
dec : List.Vector Bool n → Γ
S : Finset (Λ' Γ Λ σ)
f✝ : Γ → Stmt Bool (Λ' Γ Λ σ) σ
this :
  ∀ (i : ℕ) (f : List.Vector Bool i → Stmt Bool (Λ' Γ Λ σ) σ),
    (∀ (v : List.Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f)
hf : ∀ (a : Γ), SupportsStmt S (f✝ a)
v✝ : List.Vector Bool n
⊢ SupportsStmt S (move n Dir.left (f✝ (dec v✝)))

Proof state:
case succ
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
dec : List.Vector Bool n → Γ
S : Finset (Λ' Γ Λ σ)
f✝ : Γ → Stmt Bool (Λ' Γ Λ σ) σ
i : ℕ
IH :
  ∀ (f : List.Vector Bool i → Stmt Bool (Λ' Γ Λ σ) σ),
    (∀ (v : List.Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f)
f : List.Vector Bool (i + 1) → Stmt Bool (Λ' Γ Λ σ) σ
hf : ∀ (v : List.Vector Bool (i + 1)), SupportsStmt S (f v)
⊢ SupportsStmt S (readAux (i + 1) f)

Tactic:
constructor <;> apply IH <;> intro <;> apply hf

Result:
no goals

Proof state:
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
dec : List.Vector Bool n → Γ
S : Finset (Λ' Γ Λ σ)
f✝ : Γ → Stmt Bool (Λ' Γ Λ σ) σ
this :
  ∀ (i : ℕ) (f : List.Vector Bool i → Stmt Bool (Λ' Γ Λ σ) σ),
    (∀ (v : List.Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f)
hf : ∀ (a : Γ), SupportsStmt S (f✝ a)
v✝ : List.Vector Bool n
⊢ SupportsStmt S (move n Dir.left (f✝ (dec v✝)))

Tactic:
simp only [supportsStmt_move, hf]

Result:
no goals

Proof state:
case zero
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
dec : List.Vector Bool n → Γ
S : Finset (Λ' Γ Λ σ)
f✝ : Γ → Stmt Bool (Λ' Γ Λ σ) σ
f : List.Vector Bool 0 → Stmt Bool (Λ' Γ Λ σ) σ
hf : ∀ (v : List.Vector Bool 0), SupportsStmt S (f v)
⊢ SupportsStmt S (readAux 0 f)

Tactic:
exact hf _

Result:
no goals

Proof state:
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
dec : List.Vector Bool n → Γ
S : Finset (Λ' Γ Λ σ)
f✝ : Γ → Stmt Bool (Λ' Γ Λ σ) σ
i : ℕ
f : List.Vector Bool i → Stmt Bool (Λ' Γ Λ σ) σ
hf : ∀ (v : List.Vector Bool i), SupportsStmt S (f v)
⊢ SupportsStmt S (readAux i f)

Tactic:
induction' i with i IH

Result:
case zero
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
dec : List.Vector Bool n → Γ
S : Finset (Λ' Γ Λ σ)
f✝ : Γ → Stmt Bool (Λ' Γ Λ σ) σ
f : List.Vector Bool 0 → Stmt Bool (Λ' Γ Λ σ) σ
hf : ∀ (v : List.Vector Bool 0), SupportsStmt S (f v)
⊢ SupportsStmt S (readAux 0 f)

case succ
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
dec : List.Vector Bool n → Γ
S : Finset (Λ' Γ Λ σ)
f✝ : Γ → Stmt Bool (Λ' Γ Λ σ) σ
i : ℕ
IH :
  ∀ (f : List.Vector Bool i → Stmt Bool (Λ' Γ Λ σ) σ),
    (∀ (v : List.Vector Bool i), SupportsStmt S (f v)) → SupportsStmt S (readAux i f)
f : List.Vector Bool (i + 1) → Stmt Bool (Λ' Γ Λ σ) σ
hf : ∀ (v : List.Vector Bool (i + 1)), SupportsStmt S (f v)
⊢ SupportsStmt S (readAux (i + 1) f)