Module: Mathlib.Computability.TuringMachine.ToPartrec

Goal state before:
pp: a b : Λ'
⊢ Decidable (a = b)

Tactic:
induction a generalizing b <;> cases b

Goal state after:
case move.move
p✝¹ : Γ' → Bool
k₁✝¹ k₂✝¹ : K'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
⊢ Decidable (move p✝¹ k₁✝¹ k₂✝¹ q✝¹ = move p✝ k₁✝ k₂✝ q✝)

case move.clear
p✝¹ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
p✝ : Γ' → Bool
k✝ : K'
q✝ : Λ'
⊢ Decidable (move p✝¹ k₁✝ k₂✝ q✝¹ = clear p✝ k✝ q✝)

case move.copy
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
q✝ : Λ'
⊢ Decidable (move p✝ k₁✝ k₂✝ q✝¹ = q✝.copy)

case move.push
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
⊢ Decidable (move p✝ k₁✝ k₂✝ q✝¹ = push k✝ s✝ q✝)

case move.read
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
f✝ : Option Γ' → Λ'
⊢ Decidable (move p✝ k₁✝ k₂✝ q✝ = read f✝)

case move.succ
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
q✝ : Λ'
⊢ Decidable (move p✝ k₁✝ k₂✝ q✝¹ = q✝.succ)

case move.pred
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
q₁✝ q₂✝ : Λ'
⊢ Decidable (move p✝ k₁✝ k₂✝ q✝ = q₁✝.pred q₂✝)

case move.ret
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
k✝ : Cont'
⊢ Decidable (move p✝ k₁✝ k₂✝ q✝ = ret k✝)

case clear.move
p✝¹ : Γ' → Bool
k✝ : K'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
⊢ Decidable (clear p✝¹ k✝ q✝¹ = move p✝ k₁✝ k₂✝ q✝)

case clear.clear
p✝¹ : Γ' → Bool
k✝¹ : K'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
p✝ : Γ' → Bool
k✝ : K'
q✝ : Λ'
⊢ Decidable (clear p✝¹ k✝¹ q✝¹ = clear p✝ k✝ q✝)

case clear.copy
p✝ : Γ' → Bool
k✝ : K'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
q✝ : Λ'
⊢ Decidable (clear p✝ k✝ q✝¹ = q✝.copy)

case clear.push
p✝ : Γ' → Bool
k✝¹ : K'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
⊢ Decidable (clear p✝ k✝¹ q✝¹ = push k✝ s✝ q✝)

case clear.read
p✝ : Γ' → Bool
k✝ : K'
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
f✝ : Option Γ' → Λ'
⊢ Decidable (clear p✝ k✝ q✝ = read f✝)

case clear.succ
p✝ : Γ' → Bool
k✝ : K'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
q✝ : Λ'
⊢ Decidable (clear p✝ k✝ q✝¹ = q✝.succ)

case clear.pred
p✝ : Γ' → Bool
k✝ : K'
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
q₁✝ q₂✝ : Λ'
⊢ Decidable (clear p✝ k✝ q✝ = q₁✝.pred q₂✝)

case clear.ret
p✝ : Γ' → Bool
k✝¹ : K'
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
k✝ : Cont'
⊢ Decidable (clear p✝ k✝¹ q✝ = ret k✝)

case copy.move
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
⊢ Decidable (q✝¹.copy = move p✝ k₁✝ k₂✝ q✝)

case copy.clear
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
p✝ : Γ' → Bool
k✝ : K'
q✝ : Λ'
⊢ Decidable (q✝¹.copy = clear p✝ k✝ q✝)

case copy.copy
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
q✝ : Λ'
⊢ Decidable (q✝¹.copy = q✝.copy)

case copy.push
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
⊢ Decidable (q✝¹.copy = push k✝ s✝ q✝)

case copy.read
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
f✝ : Option Γ' → Λ'
⊢ Decidable (q✝.copy = read f✝)

case copy.succ
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
q✝ : Λ'
⊢ Decidable (q✝¹.copy = q✝.succ)

case copy.pred
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
q₁✝ q₂✝ : Λ'
⊢ Decidable (q✝.copy = q₁✝.pred q₂✝)

case copy.ret
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
k✝ : Cont'
⊢ Decidable (q✝.copy = ret k✝)

case push.move
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
⊢ Decidable (push k✝ s✝ q✝¹ = move p✝ k₁✝ k₂✝ q✝)

case push.clear
k✝¹ : K'
s✝ : Option Γ' → Option Γ'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
p✝ : Γ' → Bool
k✝ : K'
q✝ : Λ'
⊢ Decidable (push k✝¹ s✝ q✝¹ = clear p✝ k✝ q✝)

case push.copy
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
q✝ : Λ'
⊢ Decidable (push k✝ s✝ q✝¹ = q✝.copy)

case push.push
k✝¹ : K'
s✝¹ : Option Γ' → Option Γ'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
⊢ Decidable (push k✝¹ s✝¹ q✝¹ = push k✝ s✝ q✝)

case push.read
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
f✝ : Option Γ' → Λ'
⊢ Decidable (push k✝ s✝ q✝ = read f✝)

case push.succ
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
q✝ : Λ'
⊢ Decidable (push k✝ s✝ q✝¹ = q✝.succ)

case push.pred
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
q₁✝ q₂✝ : Λ'
⊢ Decidable (push k✝ s✝ q✝ = q₁✝.pred q₂✝)

case push.ret
k✝¹ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
k✝ : Cont'
⊢ Decidable (push k✝¹ s✝ q✝ = ret k✝)

case read.move
f✝ : Option Γ' → Λ'
f_ih✝ : (a : Option Γ') → (b : Λ') → Decidable (f✝ a = b)
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
⊢ Decidable (read f✝ = move p✝ k₁✝ k₂✝ q✝)

case read.clear
f✝ : Option Γ' → Λ'
f_ih✝ : (a : Option Γ') → (b : Λ') → Decidable (f✝ a = b)
p✝ : Γ' → Bool
k✝ : K'
q✝ : Λ'
⊢ Decidable (read f✝ = clear p✝ k✝ q✝)

case read.copy
f✝ : Option Γ' → Λ'
f_ih✝ : (a : Option Γ') → (b : Λ') → Decidable (f✝ a = b)
q✝ : Λ'
⊢ Decidable (read f✝ = q✝.copy)

case read.push
f✝ : Option Γ' → Λ'
f_ih✝ : (a : Option Γ') → (b : Λ') → Decidable (f✝ a = b)
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
⊢ Decidable (read f✝ = push k✝ s✝ q✝)

case read.read
f✝¹ : Option Γ' → Λ'
f_ih✝ : (a : Option Γ') → (b : Λ') → Decidable (f✝¹ a = b)
f✝ : Option Γ' → Λ'
⊢ Decidable (read f✝¹ = read f✝)

case read.succ
f✝ : Option Γ' → Λ'
f_ih✝ : (a : Option Γ') → (b : Λ') → Decidable (f✝ a = b)
q✝ : Λ'
⊢ Decidable (read f✝ = q✝.succ)

case read.pred
f✝ : Option Γ' → Λ'
f_ih✝ : (a : Option Γ') → (b : Λ') → Decidable (f✝ a = b)
q₁✝ q₂✝ : Λ'
⊢ Decidable (read f✝ = q₁✝.pred q₂✝)

case read.ret
f✝ : Option Γ' → Λ'
f_ih✝ : (a : Option Γ') → (b : Λ') → Decidable (f✝ a = b)
k✝ : Cont'
⊢ Decidable (read f✝ = ret k✝)

case succ.move
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
⊢ Decidable (q✝¹.succ = move p✝ k₁✝ k₂✝ q✝)

case succ.clear
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
p✝ : Γ' → Bool
k✝ : K'
q✝ : Λ'
⊢ Decidable (q✝¹.succ = clear p✝ k✝ q✝)

case succ.copy
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
q✝ : Λ'
⊢ Decidable (q✝¹.succ = q✝.copy)

case succ.push
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
⊢ Decidable (q✝¹.succ = push k✝ s✝ q✝)

case succ.read
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
f✝ : Option Γ' → Λ'
⊢ Decidable (q✝.succ = read f✝)

case succ.succ
q✝¹ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝¹ = b)
q✝ : Λ'
⊢ Decidable (q✝¹.succ = q✝.succ)

case succ.pred
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
q₁✝ q₂✝ : Λ'
⊢ Decidable (q✝.succ = q₁✝.pred q₂✝)

case succ.ret
q✝ : Λ'
q_ih✝ : (b : Λ') → Decidable (q✝ = b)
k✝ : Cont'
⊢ Decidable (q✝.succ = ret k✝)

case pred.move
q₁✝ q₂✝ : Λ'
q₁_ih✝ : (b : Λ') → Decidable (q₁✝ = b)
q₂_ih✝ : (b : Λ') → Decidable (q₂✝ = b)
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
⊢ Decidable (q₁✝.pred q₂✝ = move p✝ k₁✝ k₂✝ q✝)

case pred.clear
q₁✝ q₂✝ : Λ'
q₁_ih✝ : (b : Λ') → Decidable (q₁✝ = b)
q₂_ih✝ : (b : Λ') → Decidable (q₂✝ = b)
p✝ : Γ' → Bool
k✝ : K'
q✝ : Λ'
⊢ Decidable (q₁✝.pred q₂✝ = clear p✝ k✝ q✝)

case pred.copy
q₁✝ q₂✝ : Λ'
q₁_ih✝ : (b : Λ') → Decidable (q₁✝ = b)
q₂_ih✝ : (b : Λ') → Decidable (q₂✝ = b)
q✝ : Λ'
⊢ Decidable (q₁✝.pred q₂✝ = q✝.copy)

case pred.push
q₁✝ q₂✝ : Λ'
q₁_ih✝ : (b : Λ') → Decidable (q₁✝ = b)
q₂_ih✝ : (b : Λ') → Decidable (q₂✝ = b)
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
⊢ Decidable (q₁✝.pred q₂✝ = push k✝ s✝ q✝)

case pred.read
q₁✝ q₂✝ : Λ'
q₁_ih✝ : (b : Λ') → Decidable (q₁✝ = b)
q₂_ih✝ : (b : Λ') → Decidable (q₂✝ = b)
f✝ : Option Γ' → Λ'
⊢ Decidable (q₁✝.pred q₂✝ = read f✝)

case pred.succ
q₁✝ q₂✝ : Λ'
q₁_ih✝ : (b : Λ') → Decidable (q₁✝ = b)
q₂_ih✝ : (b : Λ') → Decidable (q₂✝ = b)
q✝ : Λ'
⊢ Decidable (q₁✝.pred q₂✝ = q✝.succ)

case pred.pred
q₁✝¹ q₂✝¹ : Λ'
q₁_ih✝ : (b : Λ') → Decidable (q₁✝¹ = b)
q₂_ih✝ : (b : Λ') → Decidable (q₂✝¹ = b)
q₁✝ q₂✝ : Λ'
⊢ Decidable (q₁✝¹.pred q₂✝¹ = q₁✝.pred q₂✝)

case pred.ret
q₁✝ q₂✝ : Λ'
q₁_ih✝ : (b : Λ') → Decidable (q₁✝ = b)
q₂_ih✝ : (b : Λ') → Decidable (q₂✝ = b)
k✝ : Cont'
⊢ Decidable (q₁✝.pred q₂✝ = ret k✝)

case ret.move
k✝ : Cont'
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q✝ : Λ'
⊢ Decidable (ret k✝ = move p✝ k₁✝ k₂✝ q✝)

case ret.clear
k✝¹ : Cont'
p✝ : Γ' → Bool
k✝ : K'
q✝ : Λ'
⊢ Decidable (ret k✝¹ = clear p✝ k✝ q✝)

case ret.copy
k✝ : Cont'
q✝ : Λ'
⊢ Decidable (ret k✝ = q✝.copy)

case ret.push
k✝¹ : Cont'
k✝ : K'
s✝ : Option Γ' → Option Γ'
q✝ : Λ'
⊢ Decidable (ret k✝¹ = push k✝ s✝ q✝)

case ret.read
k✝ : Cont'
f✝ : Option Γ' → Λ'
⊢ Decidable (ret k✝ = read f✝)

case ret.succ
k✝ : Cont'
q✝ : Λ'
⊢ Decidable (ret k✝ = q✝.succ)

case ret.pred
k✝ : Cont'
q₁✝ q₂✝ : Λ'
⊢ Decidable (ret k✝ = q₁✝.pred q₂✝)

case ret.ret
k✝¹ k✝ : Cont'
⊢ Decidable (ret k✝¹ = ret k✝)