| 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 |
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| end |
| end |
| end |
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| end |
| end |
| end |
| end |
|
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| end OAI |
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|
