-- Mathlib/Computability/TuringMachine.lean
theorem tr_respects_aux₂ [DecidableEq K] {k : K} {q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ} {v : σ}
    {S : ∀ k, List (Γ k)} {L : ListBlank (∀ k, Option (Γ k))}
    (hL : ∀ k, L.map (proj k) = ListBlank.mk ((S k).map some).reverse) (o : StAct K Γ σ k) :
    let v' := stVar v (S k) o
    let Sk' := stWrite v (S k) o
    let S' := update S k Sk'
    ∃ L' : ListBlank (∀ k, Option (Γ k)),
      (∀ k, L'.map (proj k) = ListBlank.mk ((S' k).map some).reverse) ∧
        TM1.stepAux (trStAct q o) v
            ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom L))) =
          TM1.stepAux q v' ((Tape.move Dir.right)^[(S' k).length] (Tape.mk' ∅ (addBottom L')))

Proof state:
case pos
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Γ k
this :
  ∀ (L R : ListBlank (Γ' K Γ)) (n : ℕ),
    Tape.write ((R.nth n).1, update (R.nth n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =
      (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R))
k' : K
i : ℕ
h' : k' = k
⊢ (if i = (S k).length then update (L.nth i) k (some (f v)) else L.nth i) k' =
    (ListBlank.mk (List.map some (update S k (f v :: S k) k')).reverse).nth i

Tactic:
subst k'

Result:
case pos
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Γ k
this :
  ∀ (L R : ListBlank (Γ' K Γ)) (n : ℕ),
    Tape.write ((R.nth n).1, update (R.nth n).2 k (some (f v))) ((Tape.move Dir.right)^[n] (Tape.mk' L R)) =
      (Tape.move Dir.right)^[n] (Tape.mk' L (ListBlank.modifyNth (fun a => (a.1, update a.2 k (some (f v)))) n R))
i : ℕ
⊢ (if i = (S k).length then update (L.nth i) k (some (f v)) else L.nth i) k =
    (ListBlank.mk (List.map some (update S k (f v :: S k) k)).reverse).nth i

Proof state:
case right.e_a.e_a
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Option (Γ k) → σ
⊢ (Tape.move Dir.left ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom L)))).head.2 k = (S k).head?

Tactic:
cases e : S k

Result:
case right.e_a.e_a.nil
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Option (Γ k) → σ
e : S k = []
⊢ (Tape.move Dir.left ((Tape.move Dir.right)^[[].length] (Tape.mk' ∅ (addBottom L)))).head.2 k = [].head?

case right.e_a.e_a.cons
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Option (Γ k) → σ
head✝ : Γ k
tail✝ : List (Γ k)
e : S k = head✝ :: tail✝
⊢ (Tape.move Dir.left ((Tape.move Dir.right)^[(head✝ :: tail✝).length] (Tape.mk' ∅ (addBottom L)))).head.2 k =
    (head✝ :: tail✝).head?

Proof state:
case right.e_a.e_a.nil
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Option (Γ k) → σ
e : S k = []
⊢ (Tape.move Dir.left ((Tape.move Dir.right)^[[].length] (Tape.mk' ∅ (addBottom L)))).head.2 k = [].head?

Tactic:
rfl

Result:
no goals