Module: Mathlib.Computability.TuringMachine.Config

Goal state before:
pp: case rfind.mp
n✝¹ : ℕ
f✝ : List.Vector ℕ n✝¹ →. ℕ
n✝ : ℕ
f : List.Vector ℕ (n✝ + 1) → ℕ
a✝ : Nat.Partrec' ↑f
cf : Code
v : List.Vector ℕ n✝
hf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]
v' v₀ : List ℕ
n : ℕ
h2 :
  v' ∈
    PFun.fix
      (fun v ↦
        (cf.eval v).bind fun y ↦
          Part.some (if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail) else Sum.inr (v.headI.succ :: v.tail)))
      (n :: ↑v)
IH :
  ∀ (a'' : List ℕ),
    (Sum.inr a'' ∈
        (cf.eval (n :: ↑v)).bind fun y ↦
          Part.some
            (if y.headI = 0 then Sum.inl ((n :: ↑v).headI.succ :: (n :: ↑v).tail)
            else Sum.inr ((n :: ↑v).headI.succ :: (n :: ↑v).tail))) →
      ∀ (n : ℕ),
        a'' = n :: ↑v →
          (∀ m < n, ¬f (m ::ᵥ v) = 0) →
            ∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ [a] = [v'.headI.pred]
hm : ∀ m < n, ¬f (m ::ᵥ v) = 0
this :
  (Sum.inl v' ∈
      (cf.eval (n :: ↑v)).bind fun y ↦
        Part.some
          (if y.headI = 0 then Sum.inl ((n :: ↑v).headI.succ :: (n :: ↑v).tail)
          else Sum.inr ((n :: ↑v).headI.succ :: (n :: ↑v).tail))) ∨
    ∃ a',
      (Sum.inr a' ∈
          (cf.eval (n :: ↑v)).bind fun y ↦
            Part.some
              (if y.headI = 0 then Sum.inl ((n :: ↑v).headI.succ :: (n :: ↑v).tail)
              else Sum.inr ((n :: ↑v).headI.succ :: (n :: ↑v).tail))) ∧
        v' ∈
          PFun.fix
            (fun v ↦
              (cf.eval v).bind fun y ↦
                Part.some (if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail) else Sum.inr (v.headI.succ :: v.tail)))
            a'
⊢ ∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ [a] = [v'.headI.pred]

Tactic:
simp only [hf, Part.bind_some] at this

Goal state after:
case rfind.mp
n✝¹ : ℕ
f✝ : List.Vector ℕ n✝¹ →. ℕ
n✝ : ℕ
f : List.Vector ℕ (n✝ + 1) → ℕ
a✝ : Nat.Partrec' ↑f
cf : Code
v : List.Vector ℕ n✝
hf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]
v' v₀ : List ℕ
n : ℕ
h2 :
  v' ∈
    PFun.fix
      (fun v ↦
        (cf.eval v).bind fun y ↦
          Part.some (if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail) else Sum.inr (v.headI.succ :: v.tail)))
      (n :: ↑v)
IH :
  ∀ (a'' : List ℕ),
    (Sum.inr a'' ∈
        (cf.eval (n :: ↑v)).bind fun y ↦
          Part.some
            (if y.headI = 0 then Sum.inl ((n :: ↑v).headI.succ :: (n :: ↑v).tail)
            else Sum.inr ((n :: ↑v).headI.succ :: (n :: ↑v).tail))) →
      ∀ (n : ℕ),
        a'' = n :: ↑v →
          (∀ m < n, ¬f (m ::ᵥ v) = 0) →
            ∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ [a] = [v'.headI.pred]
hm : ∀ m < n, ¬f (m ::ᵥ v) = 0
this :
  Sum.inl v' ∈
      Part.some
        (if [f (n ::ᵥ v)].headI = 0 then Sum.inl ((n :: ↑v).headI.succ :: (n :: ↑v).tail)
        else Sum.inr ((n :: ↑v).headI.succ :: (n :: ↑v).tail)) ∨
    ∃ a',
      Sum.inr a' ∈
          Part.some
            (if [f (n ::ᵥ v)].headI = 0 then Sum.inl ((n :: ↑v).headI.succ :: (n :: ↑v).tail)
            else Sum.inr ((n :: ↑v).headI.succ :: (n :: ↑v).tail)) ∧
        v' ∈
          PFun.fix
            (fun v ↦
              (cf.eval v).bind fun y ↦
                Part.some (if y.headI = 0 then Sum.inl (v.headI.succ :: v.tail) else Sum.inr (v.headI.succ :: v.tail)))
            a'
⊢ ∃ a, (f (a ::ᵥ v) = 0 ∧ ∀ {m : ℕ}, m < a → ¬f (m ::ᵥ v) = 0) ∧ [a] = [v'.headI.pred]