File size: 2,420 Bytes
ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | import OAI.Computability.MatchingCount.ScaledState
namespace OAI
section
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
end OAI
|