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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 | import OAI.Computability.MatchingCount.GuardRoutine
namespace OAI
section
namespace BinaryMatching.StackCode
open Polynomial
def capDouble (z : ℕ×ℕ) : ℕ×ℕ := (z.1,min z.1 (2*z.2))
lemma capDouble_spec (z : ℕ×ℕ) (k : ℕ) :
(capDouble^[k] z).1=z.1 ∧ (capDouble^[k] z).2≤ max z.1 z.2 := by
induction k with
| zero => exact ⟨rfl,Nat.le_max_right _ _⟩
| succ k ih =>
rw [Function.iterate_succ_apply']
exact ⟨ih.1,(min_le_left _ _).trans (by rw [ih.1]; exact Nat.le_max_left _ _)⟩
lemma capDouble_start (b k : ℕ) : (capDouble^[k] (b,min b 1))=(b,min b (2^k)) := by
induction k with
| zero => simp
| succ k ih =>
rw [Function.iterate_succ_apply',ih]
change (b,min b (2*min b (2^k)))=(b,min b (2^(k+1)))
rw [pow_succ]
congr 1
omega
def cappedPow2 (b k : ℕ) : ℕ := (capDouble^[k] (b,min b 1)).2
lemma cappedPow2_eq (b k : ℕ) : cappedPow2 b k=min b (2^k) := by
simp only [cappedPow2,capDouble_start]
lemma cappedPow2_of_le (b k : ℕ) (h : 2^k≤b) : cappedPow2 b k=2^k := by
rw [cappedPow2_eq,min_eq_right h]
namespace Poly
@[fun_prop] theorem capDouble_poly : Poly capDouble := by unfold capDouble; fun_prop
@[fun_prop] theorem cappedPow2_poly : Poly (fun z : ℕ×ℕ => cappedPow2 z.1 z.2) := by
have hi : Poly (fun z : ℕ×(ℕ×ℕ) => capDouble^[z.1] z.2) :=
boundedIterate capDouble capDouble_poly (3*X+1) (by
intro z j hj
obtain ⟨h₁,h₂⟩ := capDouble_spec z.2 j
rw [code_pair_length,h₁,code_nat_length,code_nat_length]
have hm : max z.2.1 z.2.2≤(encode z).length := by
simp only [code_pair_length,code_nat_length]; omega
have hn : z.2.1≤(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]
omega)
unfold cappedPow2
fun_prop
end Poly
end BinaryMatching.StackCode
end
end OAI
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