| import OAI.Computability.MatchingCount.ScaledState |
|
|
| namespace OAI |
|
|
| section |
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|
| namespace BinaryMatching.StackCode |
| open Polynomial |
|
|
| def guardedStep {α : Type} [Coding α] [Inhabited α] (f : α → α) (z : ℕ×α) : ℕ×α := |
| (z.1,if (encode (f z.2)).length≤z.1 then f z.2 else z.2) |
|
|
| lemma guardedStep_size {α : Type} [Coding α] [Inhabited α] (f : α → α) (z : ℕ×α) (j : ℕ) : |
| ((guardedStep f)^[j] z).1=z.1 ∧ |
| (encode ((guardedStep f)^[j] z).2).length≤ max z.1 (encode z.2).length := by |
| induction j with |
| | zero => exact ⟨rfl,Nat.le_max_right _ _⟩ |
| | succ j ih => |
| rw [Function.iterate_succ_apply'] |
| constructor |
| · exact ih.1 |
| · simp only [guardedStep,ih.1] |
| split_ifs with h |
| · exact h.trans (Nat.le_max_left _ _) |
| · exact ih.2 |
|
|
| lemma guardedStep_eq {α : Type} [Coding α] [Inhabited α] (f : α → α) (B : ℕ) (a : α) (k : ℕ) |
| (h : ∀ j,j≤k → (encode (f^[j] a)).length≤B) : |
| (guardedStep f)^[k] (B,a)=(B,f^[k] a) := by |
| induction k with |
| | zero => rfl |
| | succ k ih => |
| rw [Function.iterate_succ_apply',ih (fun j hj => h j (by omega))] |
| have hh := h (k+1) le_rfl |
| rw [Function.iterate_succ_apply'] at hh |
| simp only [guardedStep,ite_eq_left hh,Function.iterate_succ_apply'] |
|
|
| namespace Poly |
| variable {α : Type} [Coding α] [Inhabited α] |
| @[fun_prop] theorem guardedStep_poly (f : α → α) (hf : Poly f) : Poly (guardedStep f) := by |
| unfold guardedStep |
| have hc : Poly (fun z : ℕ×α => (encode (f z.2)).length) := |
| Poly.comp _ (fun z : ℕ×α => f z.2) Poly.codeLength (by fun_prop) |
| have hh : Poly (fun z : ℕ×α => (z.1,if decide ((encode (f z.2)).length≤z.1) then f z.2 else z.2)) := by fun_prop |
| exact hh.congr (fun _ => by simp) |
|
|
| @[fun_prop] theorem guardedIterate (f : α → α) (hf : Poly f) : |
| Poly (fun z : ℕ×(ℕ×α) => ((guardedStep f)^[z.1] z.2).2) := by |
| apply second |
| apply boundedIterate (guardedStep f) (guardedStep_poly f hf) (3*X+1) |
| intro z j hj |
| have hh := guardedStep_size f z.2 j |
| rw [code_pair_length,hh.1,code_nat_length] |
| have hb : max z.2.1 (encode z.2.2).length≤(encode z).length := by |
| simp only [code_pair_length,code_nat_length] |
| omega |
| simp only [eval_add,eval_mul,eval_ofNat,eval_X,eval_one] |
| have hN : z.2.1≤(encode z).length := by simp only [code_pair_length,code_nat_length]; omega |
| omega |
| end Poly |
| end BinaryMatching.StackCode |
| end |
|
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| end OAI |
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|
