-- Mathlib/Computability/TMToPartrec.lean
theorem move_ok {p k₁ k₂ q s L₁ o L₂} {S : K' → List Γ'} (h₁ : k₁ ≠ k₂)
    (e : splitAtPred p (S k₁) = (L₁, o, L₂)) :
    Reaches₁ (TM2.step tr) ⟨some (Λ'.move p k₁ k₂ q), s, S⟩
      ⟨some q, o, update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂))⟩

Proof state:
case cons
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
a : Γ'
L₁ : List Γ'
IH :
  ∀ {s : Option Γ'} {S : K' → List Γ'},
    splitAtPred p (S k₁) = (L₁, o, L₂) →
      Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
        { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂)) }
s : Option Γ'
S : K' → List Γ'
e : splitAtPred p (S k₁) = (a :: L₁, o, L₂)
⊢ TransGen (fun a b => b ∈ TM2.step tr a)
    (bif
        match (S k₁).head?, true, p with
        | some x, x_1, f => f x
        | none, y, x => y then
      { l := some q, var := (S k₁).head?, stk := update S k₁ (S k₁).tail }
    else
      { l := some (Λ'.move p k₁ k₂ q), var := (S k₁).head?,
        stk := update (update S k₁ (S k₁).tail) k₂ ((S k₁).head?.iget :: update S k₁ (S k₁).tail k₂) })
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ ((a :: L₁).reverseAux (S k₂)) }

Tactic:
rcases e₁ : S k₁ with - | ⟨a', Sk⟩ <;> rw [e₁, splitAtPred] at e

Result:
case cons.nil
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
a : Γ'
L₁ : List Γ'
IH :
  ∀ {s : Option Γ'} {S : K' → List Γ'},
    splitAtPred p (S k₁) = (L₁, o, L₂) →
      Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
        { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂)) }
s : Option Γ'
S : K' → List Γ'
e : ([], none, []) = (a :: L₁, o, L₂)
e₁ : S k₁ = []
⊢ TransGen (fun a b => b ∈ TM2.step tr a)
    (bif
        match [].head?, true, p with
        | some x, x_1, f => f x
        | none, y, x => y then
      { l := some q, var := [].head?, stk := update S k₁ [].tail }
    else
      { l := some (Λ'.move p k₁ k₂ q), var := [].head?,
        stk := update (update S k₁ [].tail) k₂ ([].head?.iget :: update S k₁ [].tail k₂) })
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ ((a :: L₁).reverseAux (S k₂)) }

case cons.cons
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
a : Γ'
L₁ : List Γ'
IH :
  ∀ {s : Option Γ'} {S : K' → List Γ'},
    splitAtPred p (S k₁) = (L₁, o, L₂) →
      Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
        { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂)) }
s : Option Γ'
S : K' → List Γ'
a' : Γ'
Sk : List Γ'
e :
  (bif p a' then ([], some a', Sk)
    else
      match splitAtPred p Sk with
      | (l₁, o, l₂) => (a' :: l₁, o, l₂)) =
    (a :: L₁, o, L₂)
e₁ : S k₁ = a' :: Sk
⊢ TransGen (fun a b => b ∈ TM2.step tr a)
    (bif
        match (a' :: Sk).head?, true, p with
        | some x, x_1, f => f x
        | none, y, x => y then
      { l := some q, var := (a' :: Sk).head?, stk := update S k₁ (a' :: Sk).tail }
    else
      { l := some (Λ'.move p k₁ k₂ q), var := (a' :: Sk).head?,
        stk := update (update S k₁ (a' :: Sk).tail) k₂ ((a' :: Sk).head?.iget :: update S k₁ (a' :: Sk).tail k₂) })
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ ((a :: L₁).reverseAux (S k₂)) }

Proof state:
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
s : Option Γ'
L₁ : List Γ'
o : Option Γ'
L₂ : List Γ'
S : K' → List Γ'
h₁ : k₁ ≠ k₂
e : splitAtPred p (S k₁) = (L₁, o, L₂)
⊢ Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂)) }

Tactic:
induction' L₁ with a L₁ IH generalizing S s

Result:
case nil
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
s : Option Γ'
S : K' → List Γ'
e : splitAtPred p (S k₁) = ([], o, L₂)
⊢ Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ ([].reverseAux (S k₂)) }

case cons
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
a : Γ'
L₁ : List Γ'
IH :
  ∀ {s : Option Γ'} {S : K' → List Γ'},
    splitAtPred p (S k₁) = (L₁, o, L₂) →
      Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
        { l := some q, var := o, stk := update (update S k₁ L₂) k₂ (L₁.reverseAux (S k₂)) }
s : Option Γ'
S : K' → List Γ'
e : splitAtPred p (S k₁) = (a :: L₁, o, L₂)
⊢ Reaches₁ (TM2.step tr) { l := some (Λ'.move p k₁ k₂ q), var := s, stk := S }
    { l := some q, var := o, stk := update (update S k₁ L₂) k₂ ((a :: L₁).reverseAux (S k₂)) }

Proof state:
case nil
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
s : Option Γ'
S : K' → List Γ'
e : splitAtPred p (S k₁) = ([], o, L₂)
⊢ ReflTransGen (fun a b => b ∈ TM2.step tr a)
    (TM2.stepAux
      (pop' k₁ (branch (fun s => s.elim true p) (goto fun x => q) (push' k₂ (goto fun x => Λ'.move p k₁ k₂ q)))) s S)
    { l := some q, var := o, stk := update S k₁ L₂ }

Tactic:
simp only [pop', TM2.stepAux]

Result:
case nil
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
s : Option Γ'
S : K' → List Γ'
e : splitAtPred p (S k₁) = ([], o, L₂)
⊢ ReflTransGen (fun a b => b ∈ TM2.step tr a)
    (bif (S k₁).head?.elim true p then { l := some q, var := (S k₁).head?, stk := update S k₁ (S k₁).tail }
    else TM2.stepAux (push' k₂ (goto fun x => Λ'.move p k₁ k₂ q)) (S k₁).head? (update S k₁ (S k₁).tail))
    { l := some q, var := o, stk := update S k₁ L₂ }

Proof state:
case nil.cons
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
s : Option Γ'
S : K' → List Γ'
a : Γ'
Sk : List Γ'
e :
  (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) =
    ([], o, L₂)
⊢ ReflTransGen (fun a b => b ∈ TM2.step tr a)
    (bif p a then { l := some q, var := some a, stk := update S k₁ Sk }
    else TM2.stepAux (push' k₂ (goto fun x => Λ'.move p k₁ k₂ q)) (some a) (update S k₁ Sk))
    { l := some q, var := o, stk := update S k₁ L₂ }

Tactic:
revert e

Result:
case nil.cons
p : Γ' → Bool
k₁ k₂ : K'
q : Λ'
o : Option Γ'
L₂ : List Γ'
h₁ : k₁ ≠ k₂
s : Option Γ'
S : K' → List Γ'
a : Γ'
Sk : List Γ'
⊢ (bif p a then ([], some a, Sk) else (a :: (splitAtPred p Sk).1, (splitAtPred p Sk).2.1, (splitAtPred p Sk).2.2)) =
      ([], o, L₂) →
    ReflTransGen (fun a b => b ∈ TM2.step tr a)
      (bif p a then { l := some q, var := some a, stk := update S k₁ Sk }
      else TM2.stepAux (push' k₂ (goto fun x => Λ'.move p k₁ k₂ q)) (some a) (update S k₁ Sk))
      { l := some q, var := o, stk := update S k₁ L₂ }