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e5a97c6 | 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 | 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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