Theorem: Turing.tr_eval_rev
File: Mathlib/Computability/TuringMachine.lean

tactic: cases' mem_eval.1 ab with ab b0
state_before: σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab : b₂ ∈ eval f₂ a₂
⊢ ∃ b₁, tr b₁ b₂ ∧ b₁ ∈ eval f₁ a₁

tactic: rcases tr_reaches_rev H aa ab with ⟨c₁, c₂, bc, cc, ac⟩
state_before: case intro
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab✝ : b₂ ∈ eval f₂ a₂
ab : Reaches f₂ a₂ b₂
b0 : f₂ b₂ = none
⊢ ∃ b₁, tr b₁ b₂ ∧ b₁ ∈ eval f₁ a₁

tactic: cases (reflTransGen_iff_eq (Option.eq_none_iff_forall_not_mem.1 b0)).1 bc
state_before: case intro.intro.intro.intro.intro
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab✝ : b₂ ∈ eval f₂ a₂
ab : Reaches f₂ a₂ b₂
b0 : f₂ b₂ = none
c₁ : σ₁
c₂ : σ₂
bc : Reaches f₂ b₂ c₂
cc : tr c₁ c₂
ac : Reaches f₁ a₁ c₁
⊢ ∃ b₁, tr b₁ b₂ ∧ b₁ ∈ eval f₁ a₁

tactic: refine' ⟨_, cc, mem_eval.2 ⟨ac, _⟩⟩
state_before: case intro.intro.intro.intro.intro.refl
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab✝ : b₂ ∈ eval f₂ a₂
ab : Reaches f₂ a₂ b₂
b0 : f₂ b₂ = none
c₁ : σ₁
ac : Reaches f₁ a₁ c₁
bc : Reaches f₂ b₂ b₂
cc : tr c₁ b₂
⊢ ∃ b₁, tr b₁ b₂ ∧ b₁ ∈ eval f₁ a₁

tactic: have := H cc
state_before: case intro.intro.intro.intro.intro.refl
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab✝ : b₂ ∈ eval f₂ a₂
ab : Reaches f₂ a₂ b₂
b0 : f₂ b₂ = none
c₁ : σ₁
ac : Reaches f₁ a₁ c₁
bc : Reaches f₂ b₂ b₂
cc : tr c₁ b₂
⊢ f₁ c₁ = none

tactic: cases' hfc : f₁ c₁ with d₁
state_before: case intro.intro.intro.intro.intro.refl
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab✝ : b₂ ∈ eval f₂ a₂
ab : Reaches f₂ a₂ b₂
b0 : f₂ b₂ = none
c₁ : σ₁
ac : Reaches f₁ a₁ c₁
bc : Reaches f₂ b₂ b₂
cc : tr c₁ b₂
this :
  match f₁ c₁ with
  | some b₁ => ∃ b₂_1, tr b₁ b₂_1 ∧ Reaches₁ f₂ b₂ b₂_1
  | none => f₂ b₂ = none
⊢ f₁ c₁ = none

tactic: rw [hfc] at this
state_before: case intro.intro.intro.intro.intro.refl.some
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab✝ : b₂ ∈ eval f₂ a₂
ab : Reaches f₂ a₂ b₂
b0 : f₂ b₂ = none
c₁ : σ₁
ac : Reaches f₁ a₁ c₁
bc : Reaches f₂ b₂ b₂
cc : tr c₁ b₂
this :
  match f₁ c₁ with
  | some b₁ => ∃ b₂_1, tr b₁ b₂_1 ∧ Reaches₁ f₂ b₂ b₂_1
  | none => f₂ b₂ = none
d₁ : σ₁
hfc : f₁ c₁ = some d₁
⊢ some d₁ = none

tactic: rcases this with ⟨d₂, _, bd⟩
state_before: case intro.intro.intro.intro.intro.refl.some
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab✝ : b₂ ∈ eval f₂ a₂
ab : Reaches f₂ a₂ b₂
b0 : f₂ b₂ = none
c₁ : σ₁
ac : Reaches f₁ a₁ c₁
bc : Reaches f₂ b₂ b₂
cc : tr c₁ b₂
d₁ : σ₁
this :
  match some d₁ with
  | some b₁ => ∃ b₂_1, tr b₁ b₂_1 ∧ Reaches₁ f₂ b₂ b₂_1
  | none => f₂ b₂ = none
hfc : f₁ c₁ = some d₁
⊢ some d₁ = none

tactic: rcases TransGen.head'_iff.1 bd with ⟨e, h, _⟩
state_before: case intro.intro.intro.intro.intro.refl.some.intro.intro
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab✝ : b₂ ∈ eval f₂ a₂
ab : Reaches f₂ a₂ b₂
b0 : f₂ b₂ = none
c₁ : σ₁
ac : Reaches f₁ a₁ c₁
bc : Reaches f₂ b₂ b₂
cc : tr c₁ b₂
d₁ : σ₁
hfc : f₁ c₁ = some d₁
d₂ : σ₂
left✝ : tr d₁ d₂
bd : Reaches₁ f₂ b₂ d₂
⊢ some d₁ = none

tactic: cases b0.symm.trans h
state_before: case intro.intro.intro.intro.intro.refl.some.intro.intro.intro.intro
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab✝ : b₂ ∈ eval f₂ a₂
ab : Reaches f₂ a₂ b₂
b0 : f₂ b₂ = none
c₁ : σ₁
ac : Reaches f₁ a₁ c₁
bc : Reaches f₂ b₂ b₂
cc : tr c₁ b₂
d₁ : σ₁
hfc : f₁ c₁ = some d₁
d₂ : σ₂
left✝ : tr d₁ d₂
bd : Reaches₁ f₂ b₂ d₂
e : σ₂
h : e ∈ f₂ b₂
right✝ : ReflTransGen (fun a b => b ∈ f₂ a) e d₂
⊢ some d₁ = none

tactic: rfl
state_before: case intro.intro.intro.intro.intro.refl.none
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
b₂ a₂ : σ₂
aa : tr a₁ a₂
ab✝ : b₂ ∈ eval f₂ a₂
ab : Reaches f₂ a₂ b₂
b0 : f₂ b₂ = none
c₁ : σ₁
ac : Reaches f₁ a₁ c₁
bc : Reaches f₂ b₂ b₂
cc : tr c₁ b₂
this :
  match f₁ c₁ with
  | some b₁ => ∃ b₂_1, tr b₁ b₂_1 ∧ Reaches₁ f₂ b₂ b₂_1
  | none => f₂ b₂ = none
hfc : f₁ c₁ = none
⊢ none = none