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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
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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