Module: Mathlib.Computability.TuringMachine.ToPartrec

Goal state before:
pp: case case
f g : Code
IHf :
  ∀ (k : Cont) (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepNormal f k v) b₂ ∧
        Reaches₁ (TM2.step tr)
          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b₂
IHg :
  ∀ (k : Cont) (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepNormal g k v) b₂ ∧
        Reaches₁ (TM2.step tr)
          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b₂
k : Cont
v : List ℕ
s s' : Option Γ'
⊢ Reaches₁ (TM2.step tr) { l := some (Λ'.pred ?m.739 ?m.740), var := s, stk := elim (trList v) [] ?m.741 ?m.742 }
      (Nat.rec { l := some ?m.739, var := s', stk := elim (trList v.tail) [] ?m.741 ?m.742 }
        (fun n x ↦ { l := some ?m.740, var := s', stk := elim (trList (n :: v.tail)) [] ?m.741 ?m.742 }) v.headI) →
    ∃ b₂,
      TrCfg (Nat.rec (stepNormal f k v.tail) (fun y x ↦ stepNormal g k (y :: v.tail)) v.headI) b₂ ∧
        Reaches₁ (TM2.step tr)
          { l := some (trNormal (f.case g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b₂

Tactic:
rcases v.headI with - | n <;> intro h

Goal state after:
case case.zero
f g : Code
IHf :
  ∀ (k : Cont) (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepNormal f k v) b₂ ∧
        Reaches₁ (TM2.step tr)
          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b₂
IHg :
  ∀ (k : Cont) (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepNormal g k v) b₂ ∧
        Reaches₁ (TM2.step tr)
          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b₂
k : Cont
v : List ℕ
s s' : Option Γ'
h :
  Reaches₁ (TM2.step tr) { l := some (Λ'.pred ?m.739 ?m.740), var := s, stk := elim (trList v) [] ?m.741 ?m.742 }
    (Nat.rec { l := some ?m.739, var := s', stk := elim (trList v.tail) [] ?m.741 ?m.742 }
      (fun n x ↦ { l := some ?m.740, var := s', stk := elim (trList (n :: v.tail)) [] ?m.741 ?m.742 }) 0)
⊢ ∃ b₂,
    TrCfg (Nat.rec (stepNormal f k v.tail) (fun y x ↦ stepNormal g k (y :: v.tail)) 0) b₂ ∧
      Reaches₁ (TM2.step tr)
        { l := some (trNormal (f.case g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b₂

case case.succ
f g : Code
IHf :
  ∀ (k : Cont) (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepNormal f k v) b₂ ∧
        Reaches₁ (TM2.step tr)
          { l := some (trNormal f (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b₂
IHg :
  ∀ (k : Cont) (v : List ℕ) (s : Option Γ'),
    ∃ b₂,
      TrCfg (stepNormal g k v) b₂ ∧
        Reaches₁ (TM2.step tr)
          { l := some (trNormal g (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b₂
k : Cont
v : List ℕ
s s' : Option Γ'
n : ℕ
h :
  Reaches₁ (TM2.step tr) { l := some (Λ'.pred ?m.739 ?m.740), var := s, stk := elim (trList v) [] ?m.741 ?m.742 }
    (Nat.rec { l := some ?m.739, var := s', stk := elim (trList v.tail) [] ?m.741 ?m.742 }
      (fun n x ↦ { l := some ?m.740, var := s', stk := elim (trList (n :: v.tail)) [] ?m.741 ?m.742 }) (n + 1))
⊢ ∃ b₂,
    TrCfg (Nat.rec (stepNormal f k v.tail) (fun y x ↦ stepNormal g k (y :: v.tail)) (n + 1)) b₂ ∧
      Reaches₁ (TM2.step tr)
        { l := some (trNormal (f.case g) (trCont k)), var := s, stk := elim (trList v) [] [] (trContStack k) } b₂