| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Machines.TreeFiniteOps |
| import OAI.Combinatorics.MatchingCount.Complexity.TreeNumericCosts |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.TreeTyped |
| open Polynomial |
|
|
| def fastChoose : Realizer (fun x : ℕ×ℕ=>x.1.choose x.2) := |
| (choose (composition (pair second first) leNat) natChoose (constant 0)).congr (by |
| intro x |
| simp only [Function.comp_apply,decide_eq_true_eq] |
| split_ifs with h |
| · rfl |
| · exact (Nat.choose_eq_zero_of_lt (by omega)).symm) |
| lemma fastChoose_work : WorkBound fastChoose (fun x=>x.1+weight x) := by |
| obtain ⟨p,hp⟩:=natChoose_numeric_work |
| let F := composition (pair (second (A:=ℕ) (B:=ℕ)) first) leNat |
| have hF : PolynomialTime F := polynomial_composition (polynomial_pair polynomial_second polynomial_first) polynomial_leNat |
| obtain ⟨q,hq⟩:=hF |
| obtain ⟨r,hr⟩:=polynomial_constant (A:=ℕ×ℕ) (0:ℕ) |
| refine ⟨p.comp (2*X)+q+r+7*X+18,fun x=>?_⟩ |
| have hq':=(hq x).trans (polynomial_eval_monotone q (show weight x≤x.1+weight x by omega)) |
| have hr':=(hr x).trans (polynomial_eval_monotone r (show weight x≤x.1+weight x by omega)) |
| simp only [Polynomial.eval_add,Polynomial.eval_comp,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X] |
| change F.cost x+(if decide (x.2≤x.1) then natChoose.cost x else (constant (A:=ℕ×ℕ) (0:ℕ)).cost x)+ |
| 7*weight x+18+weight (x.1.choose x.2)≤_ |
| simp only [decide_eq_true_eq] |
| by_cases h : x.2≤x.1 |
| · rw [ite_eq_left h] |
| have hp':=(hp x).trans (polynomial_eval_monotone p (show x.1+x.2≤2*(x.1+weight x) by omega)) |
| dsimp only at hq' hr' hp' |
| omega |
| · rw [ite_eq_right h,Nat.choose_eq_zero_of_lt (show x.1<x.2 by omega)] |
| dsimp only at hq' hr' |
| omega |
|
|
| lemma polynomial_boundedFastChoose {A : Type} [Codec A] {f : A → ℕ×ℕ} {F : Realizer f} |
| (hF : PolynomialTime F) (p : Polynomial ℕ) |
| (hn : ∀ a,(f a).1≤p.eval (weight a)) : PolynomialTime (composition F fastChoose) := by |
| obtain ⟨q,hq⟩:=hF |
| apply work_composition ⟨q,hq⟩ fastChoose_work (p+q) |
| intro a |
| have h1:=hn a |
| have h2:=hq a |
| simp only [Polynomial.eval_add] |
| omega |
| end MatchingFPRAS.TreeTyped |
|
|
| end OAI |
|
|
