-- Mathlib/Computability/TMToPartrec.lean
theorem succ_ok {q s n} {c d : List Γ'} :
    Reaches₁ (TM2.step tr) ⟨some (Λ'.succ q), s, K'.elim (trList [n]) [] c d⟩
      ⟨some q, none, K'.elim (trList [n.succ]) [] c d⟩

Proof state:
q : Λ'
s : Option Γ'
n : ℕ
c d : List Γ'
⊢ Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trList [n]) [] c d }
    { l := some q, var := none, stk := elim (trList [n.succ]) [] c d }

Tactic:
simp only [TM2.step, trList, trNat.eq_1, Nat.cast_succ, Num.add_one]

Result:
q : Λ'
s : Option Γ'
n : ℕ
c d : List Γ'
⊢ Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trNum ↑n ++ [Γ'.cons]) [] c d }
    { l := some q, var := none, stk := elim (trNum (↑n).succ ++ [Γ'.cons]) [] c d }

Proof state:
case intro.intro.intro.intro
q : Λ'
s : Option Γ'
n : ℕ
c d : List Γ'
a : PosNum
this :
  ∀ (l₁ : List Γ'),
    ∃ l₁' l₂' s',
      l₁.reverseAux (trPosNum a.succ) = l₁'.reverseAux l₂' ∧
        Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) l₁ c d }
          { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
l₁' l₂' : List Γ'
s' : Option Γ'
e : [].reverseAux (trPosNum a.succ) = l₁'.reverseAux l₂'
h :
  Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) [] c d }
    { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
⊢ Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) [] c d }
    { l := some q, var := none, stk := elim (trPosNum a.succ ++ [Γ'.cons]) [] c d }

Tactic:
simp? [List.reverseAux] at e says simp only [List.reverseAux, List.reverseAux_eq] at e

Result:
case intro.intro.intro.intro
q : Λ'
s : Option Γ'
n : ℕ
c d : List Γ'
a : PosNum
this :
  ∀ (l₁ : List Γ'),
    ∃ l₁' l₂' s',
      l₁.reverseAux (trPosNum a.succ) = l₁'.reverseAux l₂' ∧
        Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) l₁ c d }
          { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
l₁' l₂' : List Γ'
s' : Option Γ'
h :
  Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) [] c d }
    { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
e : trPosNum a.succ = l₁'.reverse ++ l₂'
⊢ Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) [] c d }
    { l := some q, var := none, stk := elim (trPosNum a.succ ++ [Γ'.cons]) [] c d }

Proof state:
case pos.bit1
q : Λ'
n : ℕ
c d : List Γ'
m : PosNum
IH :
  ∀ {s : Option Γ'} (l₁ : List Γ'),
    ∃ l₁' l₂' s',
      l₁.reverseAux (trPosNum m.succ) = l₁'.reverseAux l₂' ∧
        Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum m ++ [Γ'.cons]) l₁ c d }
          { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
s : Option Γ'
l₁ : List Γ'
⊢ ∃ l₁' l₂' s',
    l₁.reverseAux (trPosNum m.bit1.succ) = l₁'.reverseAux l₂' ∧
      Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum m.bit1 ++ [Γ'.cons]) l₁ c d }
        { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }

Tactic:
obtain ⟨l₁', l₂', s', e, h⟩ := IH (Γ'.bit0 :: l₁)

Result:
case pos.bit1.intro.intro.intro.intro
q : Λ'
n : ℕ
c d : List Γ'
m : PosNum
IH :
  ∀ {s : Option Γ'} (l₁ : List Γ'),
    ∃ l₁' l₂' s',
      l₁.reverseAux (trPosNum m.succ) = l₁'.reverseAux l₂' ∧
        Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum m ++ [Γ'.cons]) l₁ c d }
          { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
s : Option Γ'
l₁ l₁' l₂' : List Γ'
s' : Option Γ'
e : (Γ'.bit0 :: l₁).reverseAux (trPosNum m.succ) = l₁'.reverseAux l₂'
h :
  Reaches₁ (TM2.step tr)
    { l := some q.succ, var := ?m.108487, stk := elim (trPosNum m ++ [Γ'.cons]) (Γ'.bit0 :: l₁) c d }
    { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
⊢ ∃ l₁' l₂' s',
    l₁.reverseAux (trPosNum m.bit1.succ) = l₁'.reverseAux l₂' ∧
      Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum m.bit1 ++ [Γ'.cons]) l₁ c d }
        { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }

Proof state:
case intro.intro.intro.intro
q : Λ'
s : Option Γ'
n : ℕ
c d : List Γ'
a : PosNum
this :
  ∀ (l₁ : List Γ'),
    ∃ l₁' l₂' s',
      l₁.reverseAux (trPosNum a.succ) = l₁'.reverseAux l₂' ∧
        Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) l₁ c d }
          { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
l₁' l₂' : List Γ'
s' : Option Γ'
h :
  Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) [] c d }
    { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
e : trPosNum a.succ = l₁'.reverse ++ l₂'
⊢ Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) [] c d }
    { l := some q, var := none, stk := elim (trPosNum a.succ ++ [Γ'.cons]) [] c d }

Tactic:
refine h.trans ?_

Result:
case intro.intro.intro.intro
q : Λ'
s : Option Γ'
n : ℕ
c d : List Γ'
a : PosNum
this :
  ∀ (l₁ : List Γ'),
    ∃ l₁' l₂' s',
      l₁.reverseAux (trPosNum a.succ) = l₁'.reverseAux l₂' ∧
        Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) l₁ c d }
          { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
l₁' l₂' : List Γ'
s' : Option Γ'
h :
  Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) [] c d }
    { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
e : trPosNum a.succ = l₁'.reverse ++ l₂'
⊢ TransGen (fun a b => b ∈ TM2.step tr a) { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
    { l := some q, var := none, stk := elim (trPosNum a.succ ++ [Γ'.cons]) [] c d }

Proof state:
case pos.bit0
q : Λ'
n : ℕ
c d : List Γ'
m : PosNum
a_ih✝ :
  ∀ {s : Option Γ'} (l₁ : List Γ'),
    ∃ l₁' l₂' s',
      l₁.reverseAux (trPosNum m.succ) = l₁'.reverseAux l₂' ∧
        Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum m ++ [Γ'.cons]) l₁ c d }
          { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
s : Option Γ'
l₁ : List Γ'
⊢ { l := some (unrev q), var := some Γ'.bit0, stk := elim (trPosNum m.bit0.succ ++ [Γ'.cons]) l₁ c d } ∈
    TM2.step tr { l := some q.succ, var := s, stk := elim (trPosNum m.bit0 ++ [Γ'.cons]) l₁ c d }

Tactic:
simp only [TM2.step]

Result:
case pos.bit0
q : Λ'
n : ℕ
c d : List Γ'
m : PosNum
a_ih✝ :
  ∀ {s : Option Γ'} (l₁ : List Γ'),
    ∃ l₁' l₂' s',
      l₁.reverseAux (trPosNum m.succ) = l₁'.reverseAux l₂' ∧
        Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum m ++ [Γ'.cons]) l₁ c d }
          { l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
s : Option Γ'
l₁ : List Γ'
⊢ { l := some (unrev q), var := some Γ'.bit0, stk := elim (trPosNum m.bit0.succ ++ [Γ'.cons]) l₁ c d } ∈
    some (TM2.stepAux (tr q.succ) s (elim (trPosNum m.bit0 ++ [Γ'.cons]) l₁ c d))