| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Tapes.OnlineWord |
| import OAI.Combinatorics.MatchingCount.Complexity.OnlineFrameCost |
| import OAI.Combinatorics.MatchingCount.Complexity.OnlineFoldCost |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineWord |
| open TreeTyped OnlineTree PhysicalStream Polynomial |
|
|
| lemma polynomial_step : OnlineTree.PolynomialTime stepRealizer := by |
| unfold stepRealizer |
| apply OnlineTree.work_congr |
| apply OnlineTree.polynomial_choose <;> apply OnlineTree.polynomial_deterministic <;> |
| apply TreeTyped.polynomial_composition <;> first | exact TreeTyped.polynomial_listCons | |
| exact TreeTyped.polynomial_pair (TreeTyped.polynomial_constant _) TreeTyped.polynomial_second |
|
|
| lemma weight_bools (bs : List Bool) : weight bs≤4*bs.length+1 := by |
| induction bs with |
| | nil => simp |
| | cons b bs ih => have hb:=TreeTyped.weight_bool b; simp only [weight_cons,List.length_cons];omega |
| lemma all_weight (n : ℕ) (acc : List Bool) : |
| (prepend n acc).All (fun bs=>weight bs≤4*(n+acc.length)+1) := |
| (all_length n acc).mono (fun bs hb=>by rw [←hb]; exact weight_bools bs) |
|
|
| lemma polynomial_list : OnlineTree.PolynomialTime listRealizer := by |
| obtain ⟨p,hp⟩:=polynomial_step |
| have prep := TreeTyped.polynomial_pair (TreeTyped.polynomial_identity (A:=List ℕ)) |
| (TreeTyped.polynomial_constant ([] : List Bool)) |
| obtain ⟨q,hq⟩:=prep |
| refine ⟨q+X*((p.comp (5*X+2))+14*X+34)+4*X+11,fun as=>?_⟩ |
| let S:=weight as |
| have hlen : as.length≤S := list_length_le_weight as |
| have bound := OnlineTree.fold_cost_bound step stepRealizer.cost as ([] : List Bool) |
| (fun k bs=>bs.length+k=as.length) (p.eval (5*S+2)) S (by simp) le_rfl |
| (by |
| intro a ha k hk bs hb |
| simp only [step,Experiment.All,List.length_cons] |
| constructor <;> omega) |
| (by |
| intro a ha k hk bs hb |
| have ha' : weight a≤S := weight_mem ha |
| have hbs:=weight_bools bs |
| have hw : weight (a,bs)≤5*S+2 := by rw [weight_prod];omega |
| exact (Nat.le_add_right _ _).trans ((hp (a,bs)).trans (polynomial_eval_monotone p hw))) |
| have hlenmul := Nat.mul_le_mul_right (p.eval (5*S+2)+14*S+34) hlen |
| have hout := Experiment.worst_le _ _ _ (all_weight as.length []) |
| have hpre:=hq as |
| change (TreeTyped.pair TreeTyped.identity (TreeTyped.constant ([] : List Bool))).cost as+ |
| OnlineTree.Fold.cost step stepRealizer.cost as []+(prepend as.length []).worst weight≤_ |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_comp,Polynomial.eval_X,Polynomial.eval_ofNat] |
| dsimp [S] at * |
| omega |
|
|
| lemma work_natural : OnlineTree.WorkBound naturalRealizer (fun n=>n+weight n) := by |
| apply OnlineTree.work_congr |
| apply OnlineTree.work_then_polynomial _ polynomial_list |
| exact OnlineTree.work_deterministic (TreeTyped.work_remeasure TreeTyped.range_numeric_work X (fun n=>by simp)) |
| end MatchingFPRAS.OnlineWord |
|
|
| end OAI |
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|
