Module: Mathlib.Computability.TuringMachine.StackTuringMachine

Goal state before:
pp: K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
q : TM2.Stmt Γ Λ σ
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct K Γ σ k
IH :
  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},
    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →
      ∃ b,
        TrCfg (TM2.stepAux q v S) b ∧
          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b
hgo :
  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }
    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }
T' : ListBlank ((k : K) → Option (Γ k))
hT' :
  ∀ (k_1 : K),
    ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.149 (S k) o) k_1)).reverse
hrun :
  TM1.stepAux (trStAct ?m.148 o) ?m.149 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =
    TM1.stepAux ?m.148 (stVar ?m.149 (S k) o)
      ((Tape.move Dir.right)^[(update S k (stWrite ?m.149 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))
this :
  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }
    (bif (((Tape.mk' ∅ (addBottom T)).nth ↑(S k).length).2 k).isNone then
      TM1.stepAux (trStAct (goto fun x x_1 ↦ ret q) o) v
        ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))
    else
      TM1.stepAux (move Dir.right (goto fun x x_1 ↦ go k o q)) v
        ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))))
⊢ ∃ b,
    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧
      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x_1 ↦ go k o q) v (Tape.mk' ∅ (addBottom T))) b

Tactic:
Tape.mk'_nth_nat,

Goal state after:
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
q : TM2.Stmt Γ Λ σ
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct K Γ σ k
IH :
  ∀ {v : σ} {S : (k : K) → List (Γ k)} {T : ListBlank ((k : K) → Option (Γ k))},
    (∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse) →
      ∃ b,
        TrCfg (TM2.stepAux q v S) b ∧
          Reaches (TM1.step (tr M)) (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom T))) b
hgo :
  Reaches₀ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }
    { l := some (go k o q), var := v, Tape := (Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)) }
T' : ListBlank ((k : K) → Option (Γ k))
hT' :
  ∀ (k_1 : K),
    ListBlank.map (proj k_1) T' = ListBlank.mk (List.map some (update S k (stWrite ?m.149 (S k) o) k_1)).reverse
hrun :
  TM1.stepAux (trStAct ?m.148 o) ?m.149 ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))) =
    TM1.stepAux ?m.148 (stVar ?m.149 (S k) o)
      ((Tape.move Dir.right)^[(update S k (stWrite ?m.149 (S k) o) k).length] (Tape.mk' ∅ (addBottom T')))
this :
  Reaches₁ (TM1.step (tr M)) { l := some (go k o q), var := v, Tape := Tape.mk' ∅ (addBottom T) }
    (bif (((addBottom T).nth (S k).length).2 k).isNone then
      TM1.stepAux (trStAct (goto fun x x_1 ↦ ret q) o) v
        ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T)))
    else
      TM1.stepAux (move Dir.right (goto fun x x_1 ↦ go k o q)) v
        ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom T))))
⊢ ∃ b,
    TrCfg (TM2.stepAux q (stVar v (S k) o) (update S k (stWrite v (S k) o))) b ∧
      Reaches (TM1.step (tr M)) (TM1.stepAux (goto fun x x_1 ↦ go k o q) v (Tape.mk' ∅ (addBottom T))) b