import Mathlib
import OAI.Computability.VertexCover.Machines.Lists

namespace OAI

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namespace VertexCover.Machine
open Turing

noncomputable def Poly.loopFrom {α β : Type} {e : α → List Bool} {body : α → Bool × α}
    (c : Poly e (flagBits e) body) (initial result : β → α) (size count : Polynomial ℕ)
    (spec : ∀ b, ∃ n, n ≤ count.eval (e (initial b)).length ∧
      BoundedRun e body (size.eval (e (initial b)).length) (initial b) (result b) n) :
    Poly (fun b => e (initial b)) e result where
  computation := {
    tm := Loop.machine c.computation.tm c.computation.inputAlphabet c.computation.outputAlphabet
    inputAlphabet := c.computation.inputAlphabet
    outputAlphabet := Equiv.refl Bool
    time := count * (loopCost c.computation).comp size
    outputsFun := fun a => Classical.choice (by
      simp only [show (Equiv.refl Bool).invFun = id from rfl, List.map_id]
      obtain ⟨n, hn, run⟩ := spec a
      obtain ⟨execution⟩ := boundedRun_execution c.computation run
      refine ⟨{ execution with steps_le_m := ?_ }⟩
      simp only [Polynomial.eval_mul, Polynomial.eval_comp]
      exact execution.steps_le_m.trans (Nat.mul_le_mul_right _ hn)) }
  finiteAlphabet := Loop.finiteAlphabet _ _ _ c.finiteAlphabet

noncomputable def Poly.initializeLoop {α β : Type} {ea : α → List Bool} {eb : β → List Bool}
    {initial : α → β} {result : α → β} (ci : Poly ea eb initial)
    (cl : Poly (fun a => eb (initial a)) eb result) : Poly ea eb result := by
  let first : Poly ea (fun a => eb (initial a)) id := ci.encodeCongr id (fun _ => rfl) (fun _ => rfl)
  exact (first.comp cl).congr (fun _ => rfl)

end VertexCover.Machine
end


end
end
end
end
end
end
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end
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end
end
end

end OAI