-- Mathlib/Computability/TMConfig.lean
theorem exists_code {n} {f : List.Vector ℕ n →. ℕ} (hf : Nat.Partrec' f) :
    ∃ c : Code, ∀ v : List.Vector ℕ n, c.eval v.1 = pure <$> f v

Proof state:
case prim.prec.intro.intro.succ.succ
n✝² : ℕ
f✝¹ : List.Vector ℕ n✝² →. ℕ
n✝¹ : ℕ
f✝ : List.Vector ℕ n✝¹ → ℕ
n✝ : ℕ
f : List.Vector ℕ n✝ → ℕ
g : List.Vector ℕ (n✝ + 2) → ℕ
a✝² : Nat.Primrec' f
a✝¹ : Nat.Primrec' g
cf cg : Code
v : List.Vector ℕ (n✝ + 1)
hf : cf.eval (↑v).tail = pure (pure (f v.tail))
hg : ∀ (a b : ℕ), cg.eval (a :: b :: (↑v).tail) = pure (pure (g (a ::ᵥ b ::ᵥ v.tail)))
n : ℕ
a✝ : (cf.prec cg).eval (n :: (↑v).tail) = pure (pure (Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) n))
b : ℕ
IH :
  ∀ (a : ℕ),
    a + b = n →
      n.succ :: 0 :: g (n ::ᵥ Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) n ::ᵥ v.tail) :: (↑v).tail ∈
        PFun.fix
          (fun v =>
            (cg.eval (v.headI :: v.tail.tail)).bind fun x =>
              Part.some
                (if v.tail.headI = 0 then Sum.inl (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail)
                else Sum.inr (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail)))
          (a :: b :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail)
a : ℕ
e : a + (b + 1) = n
⊢ n.succ :: 0 :: g (n ::ᵥ Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) n ::ᵥ v.tail) :: (↑v).tail ∈
    PFun.fix
      (fun v =>
        (cg.eval (v.headI :: v.tail.tail)).bind fun x =>
          Part.some
            (if v.tail.headI = 0 then Sum.inl (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail)
            else Sum.inr (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail)))
      (a :: (b + 1) :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail)

Tactic:
refine PFun.mem_fix_iff.2 (Or.inr ⟨_, ?_, IH (a + 1) (by rwa [add_right_comm])⟩)

Result:
case prim.prec.intro.intro.succ.succ
n✝² : ℕ
f✝¹ : List.Vector ℕ n✝² →. ℕ
n✝¹ : ℕ
f✝ : List.Vector ℕ n✝¹ → ℕ
n✝ : ℕ
f : List.Vector ℕ n✝ → ℕ
g : List.Vector ℕ (n✝ + 2) → ℕ
a✝² : Nat.Primrec' f
a✝¹ : Nat.Primrec' g
cf cg : Code
v : List.Vector ℕ (n✝ + 1)
hf : cf.eval (↑v).tail = pure (pure (f v.tail))
hg : ∀ (a b : ℕ), cg.eval (a :: b :: (↑v).tail) = pure (pure (g (a ::ᵥ b ::ᵥ v.tail)))
n : ℕ
a✝ : (cf.prec cg).eval (n :: (↑v).tail) = pure (pure (Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) n))
b : ℕ
IH :
  ∀ (a : ℕ),
    a + b = n →
      n.succ :: 0 :: g (n ::ᵥ Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) n ::ᵥ v.tail) :: (↑v).tail ∈
        PFun.fix
          (fun v =>
            (cg.eval (v.headI :: v.tail.tail)).bind fun x =>
              Part.some
                (if v.tail.headI = 0 then Sum.inl (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail)
                else Sum.inr (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail)))
          (a :: b :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail)
a : ℕ
e : a + (b + 1) = n
⊢ Sum.inr ((a + 1) :: b :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) (a + 1) :: (↑v).tail) ∈
    (cg.eval
          ((a :: (b + 1) :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail).headI ::
            (a :: (b + 1) :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail).tail.tail)).bind
      fun x =>
      Part.some
        (if
            (a :: (b + 1) :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail).tail.headI =
              0 then
          Sum.inl
            ((a :: (b + 1) :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail).headI.succ ::
              (a ::
                        (b + 1) ::
                          Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail).tail.headI.pred ::
                x.headI ::
                  (a ::
                          (b + 1) ::
                            Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail).tail.tail.tail)
        else
          Sum.inr
            ((a :: (b + 1) :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail).headI.succ ::
              (a ::
                        (b + 1) ::
                          Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail).tail.headI.pred ::
                x.headI ::
                  (a ::
                          (b + 1) ::
                            Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail).tail.tail.tail))

Proof state:
case prim.zero
n : ℕ
f : List.Vector ℕ n →. ℕ
n✝ : ℕ
f✝ : List.Vector ℕ n✝ → ℕ
⊢ ∃ c, ∀ (v : List.Vector ℕ 0), c.eval ↑v = pure <$> (↑fun x => 0) v

Tactic:
exact ⟨zero', fun ⟨[], _⟩ => rfl⟩

Result:
no goals

Proof state:
n✝ : ℕ
f✝ : List.Vector ℕ n✝ →. ℕ
n : ℕ
f : List.Vector ℕ (n + 1) → ℕ
a✝ : Nat.Partrec' ↑f
cf : Code
v : List.Vector ℕ n
hf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]
v' : List ℕ
h1 :
  v' ∈
    PFun.fix
      (fun v =>
        (cf.eval v).bind fun y =>
          Part.some (if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail) else Sum.inr (v.headI.succ :: v.tail)))
      (0 :: ↑v)
this :
  ∀ (v₁ : List ℕ),
    v' ∈
        PFun.fix
          (fun v =>
            (cf.eval v).bind fun y =>
              Part.some (if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail) else Sum.inr (v.headI.succ :: v.tail)))
          v₁ →
      ∀ (n_1 : ℕ),
        v₁ = n_1 :: ↑v →
          (∀ m < n_1, ¬f (m ::ᵥ v) = 0) →
            ∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ [a] = [v'.headI.pred]
⊢ ∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ pure a = [v'.headI.pred]

Tactic:
exact this _ h1 0 rfl (by rintro _ ⟨⟩)

Result:
no goals

Proof state:
n✝² : ℕ
f✝¹ : List.Vector ℕ n✝² →. ℕ
n✝¹ : ℕ
f✝ : List.Vector ℕ n✝¹ → ℕ
n✝ : ℕ
f : List.Vector ℕ n✝ → ℕ
g : List.Vector ℕ (n✝ + 2) → ℕ
a✝² : Nat.Primrec' f
a✝¹ : Nat.Primrec' g
cf cg : Code
v : List.Vector ℕ (n✝ + 1)
hf : cf.eval (↑v).tail = pure (pure (f v.tail))
hg : ∀ (a b : ℕ), cg.eval (a :: b :: (↑v).tail) = pure (pure (g (a ::ᵥ b ::ᵥ v.tail)))
n : ℕ
a✝ : (cf.prec cg).eval (n :: (↑v).tail) = pure (pure (Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) n))
b : ℕ
IH :
  ∀ (a : ℕ),
    a + b = n →
      n.succ :: 0 :: g (n ::ᵥ Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) n ::ᵥ v.tail) :: (↑v).tail ∈
        PFun.fix
          (fun v =>
            (cg.eval (v.headI :: v.tail.tail)).bind fun x =>
              Part.some
                (if v.tail.headI = 0 then Sum.inl (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail)
                else Sum.inr (v.headI.succ :: v.tail.headI.pred :: x.headI :: v.tail.tail.tail)))
          (a :: b :: Nat.rec (f v.tail) (fun y IH => g (y ::ᵥ IH ::ᵥ v.tail)) a :: (↑v).tail)
a : ℕ
e : a + (b + 1) = n
⊢ a + 1 + b = n

Tactic:
rwa [add_right_comm]

Result:
no goals