| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineRequest |
| import OAI.Combinatorics.MatchingCount.Complexity.OnlineSampleCost |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.NumericBounds |
| open Polynomial TreeTyped |
|
|
| variable {A B : Type} |
| def Bound (f m : A→ℕ) : Prop := ∃ p : Polynomial ℕ,∀ x,f x≤p.eval (m x) |
| lemma self (m : A→ℕ) : Bound m m := ⟨X,fun _=>by simp⟩ |
| lemma of_le {f m : A→ℕ} (h : ∀ x,f x ≤ m x) : Bound f m := ⟨X,fun x=>by simpa using h x⟩ |
| lemma const (n : ℕ) (m : A→ℕ) : Bound (fun _=>n) m := ⟨C n,fun _=>by simp⟩ |
| lemma mono {f g m : A→ℕ} (hf : Bound f m) (h : ∀ x,g x≤f x) : Bound g m := by |
| obtain ⟨p,hp⟩:=hf;exact ⟨p,fun x=>(h x).trans (hp x)⟩ |
| lemma add {f g m : A→ℕ} (hf : Bound f m) (hg : Bound g m) : Bound (fun x=>f x+g x) m := by |
| obtain ⟨p,hp⟩:=hf;obtain ⟨q,hq⟩:=hg |
| exact ⟨p+q,fun x=>by simpa using Nat.add_le_add (hp x) (hq x)⟩ |
| lemma mul {f g m : A→ℕ} (hf : Bound f m) (hg : Bound g m) : Bound (fun x=>f x*g x) m := by |
| obtain ⟨p,hp⟩:=hf;obtain ⟨q,hq⟩:=hg |
| exact ⟨p*q,fun x=>by simpa using Nat.mul_le_mul (hp x) (hq x)⟩ |
| lemma pow {f m : A→ℕ} (hf : Bound f m) (n : ℕ) : Bound (fun x=>f x^n) m := by |
| obtain ⟨p,hp⟩:=hf |
| exact ⟨p^n,fun x=>by simpa using Nat.pow_le_pow_left (hp x) n⟩ |
| lemma clog_le (n : ℕ) : Nat.clog 2 n≤n := by |
| apply Nat.clog_le_of_le_pow |
| exact (Nat.lt_two_pow_self (n:=n)).le |
| lemma clog {f m : A→ℕ} (hf : Bound f m) : Bound (fun x=>Nat.clog 2 (f x)) m := |
| mono hf (fun _=>clog_le _) |
| lemma comp {f : A→B} {m : A→ℕ} {n g : B→ℕ} (hg : Bound g n) (hf : Bound (n ∘ f) m) : |
| Bound (g ∘ f) m := by |
| obtain ⟨p,hp⟩:=hg;obtain ⟨q,hq⟩:=hf |
| exact ⟨p.comp q,fun x=>by simpa using (hp (f x)).trans (polynomial_eval_monotone p (hq x))⟩ |
| lemma weight [Codec A] [Codec B] {f : A→B} {F : Realizer f} {m : A→ℕ} |
| (hF : WorkBound F m) : Bound (fun x=>TreeTyped.weight (f x)) m := by |
| obtain ⟨p,hp⟩:=hF |
| exact ⟨p,fun x=>(Nat.le_add_left _ _).trans (hp x)⟩ |
| lemma length [Codec A] [Codec B] {f : A→List B} {F : Realizer f} {m : A→ℕ} |
| (hF : WorkBound F m) : Bound (fun x=>(f x).length) m := |
| mono (weight hF) (fun _=>list_length_le_weight _) |
| end MatchingFPRAS.NumericBounds |
|
|
| namespace MatchingFPRAS.OnlineSchedule |
| open TreeTyped Polynomial NumericBounds Transport ExecutableSource |
|
|
| def precisionNorm (x : PrecisionData) : ℕ := |
| x.1.1.1.1+x.1.1.1.2+x.1.1.2.1+x.1.1.2.2+x.1.2+x.2.1+x.2.2 |
| lemma bound_plan : Bound (fun x : PrecisionData=> |
| (precisionPlan x).1.1.1+(precisionPlan x).1.1.2+(precisionPlan x).1.2+(precisionPlan x).2) precisionNorm := by |
| have hN : Bound (fun x : PrecisionData=>x.1.1.1.1) precisionNorm := of_le (fun x=>by unfold precisionNorm;omega) |
| have hD : Bound (fun x : PrecisionData=>x.1.1.1.2) precisionNorm := of_le (fun x=>by unfold precisionNorm;omega) |
| have hK : Bound (fun x : PrecisionData=>x.1.1.2.1) precisionNorm := of_le (fun x=>by unfold precisionNorm;omega) |
| have hT : Bound (fun x : PrecisionData=>x.1.1.2.2) precisionNorm := of_le (fun x=>by unfold precisionNorm;omega) |
| have hm : Bound (fun x : PrecisionData=>x.1.2) precisionNorm := of_le (fun x=>by unfold precisionNorm;omega) |
| have hE : Bound (fun x : PrecisionData=>x.2.1) precisionNorm := of_le (fun x=>by unfold precisionNorm;omega) |
| have hB : Bound (fun x : PrecisionData=>x.2.2) precisionNorm := of_le (fun x=>by unfold precisionNorm;omega) |
| have hL : Bound (fun x : PrecisionData=>ladderL x.1.1.1.1 x.1.1.1.2) precisionNorm := |
| pow (mul (mul (const (10^4) _) hN) hD) 100 |
| have hQ : Bound (fun x : PrecisionData=>ladderQ x.1.1.1.1 x.1.1.1.2) precisionNorm := mul (const 32 _) hL |
| have hM : Bound (fun x : PrecisionData=>ladderM x.1.1.1.1 x.1.1.1.2 x.1.1.2.2) precisionNorm := |
| add (mul hT (pow hQ 2)) hQ |
| have hlambda : Bound (fun x : PrecisionData=>ladderLambda x.1.1.1.1 x.1.1.1.2 x.1.1.2.1 x.1.1.2.2) precisionNorm := |
| mul (mul (mul (mul (const 10 _) hN) hQ) (add hT (const 1 _))) |
| (add (add hK (clog (add (mul hN hD) (const 2 _)))) (const 1 _)) |
| have hr : Bound (fun x : PrecisionData=>discreteLadderSteps x.1.1.1.1 x.1.1.1.2 x.1.1.2.1 x.1.1.2.2 x.1.2) precisionNorm := |
| mul (mul (mul (const 20 _) hL) hM) (add (add hlambda (clog hm)) (const 1 _)) |
| have he : Bound scanError precisionNorm := |
| add (add (mul (const 2 _) (add (const 1 _) hM)) |
| (mul (const 4 _) (add (add (mul (const 8 _) (pow hE 2)) (mul (const 4 _) hE)) (const 3 _)))) |
| (mul (mul (const 2 _) (pow hN 2)) hB) |
| exact add (add (add (clog (mul (mul hr he) hm)) hN) hr) (clog (mul (mul (const 6 _) hE) hm)) |
|
|
| def requestSize (x : RequestInput) : ℕ := x.1.1.2.1.2+x.2+TreeTyped.weight x |
| lemma weight_le_requestSize (x : RequestInput) : TreeTyped.weight x≤requestSize x := by unfold requestSize;omega |
| lemma bound_precision : Bound (precisionNorm ∘ sourcePrecision) requestSize := by |
| have hn := NumericBounds.length (work_of_polynomial |
| (polynomial_composition polynomial_requestBag polynomial_sourceVertices) X |
| (fun x=>by simpa using weight_le_requestSize x)) |
| have he := NumericBounds.length (work_of_polynomial polynomial_requestCatalog X |
| (fun x=>by simpa using weight_le_requestSize x)) |
| have hb := NumericBounds.length (work_of_polynomial |
| (polynomial_composition polynomial_requestBag polynomial_sourceParents) X |
| (fun x=>by simpa using weight_le_requestSize x)) |
| have hd : Bound (fun x : RequestInput=>x.1.1.2.1.2) requestSize := of_le (fun x=>by unfold requestSize;omega) |
| have hs : Bound (fun x : RequestInput=>x.2) requestSize := of_le (fun x=>by unfold requestSize;omega) |
| have hk : Bound (fun x : RequestInput=>x.1.1.2.1.1.2.length/2) requestSize := by |
| apply of_le;intro x |
| have h:=list_length_le_weight x.1.1.2.1.1.2 |
| have hd:=Nat.div_le_self x.1.1.2.1.1.2.length 2 |
| unfold requestSize |
| have h1:=weight_fst x |
| have h2:=weight_fst x.1 |
| have h3:=weight_snd x.1.1 |
| have h4:=weight_fst x.1.1.2 |
| have h5:=weight_fst x.1.1.2.1 |
| have h6:=weight_snd x.1.1.2.1.1 |
| omega |
| have ht : Bound (fun x : RequestInput=>x.1.1.2.2.1.length) requestSize := by |
| apply of_le;intro x |
| have h:=list_length_le_weight x.1.1.2.2.1 |
| unfold requestSize |
| have h1:=weight_fst x |
| have h2:=weight_fst x.1 |
| have h3:=weight_snd x.1.1 |
| have h7:=weight_snd x.1.1.2 |
| have h8:=weight_fst x.1.1.2.2 |
| omega |
| exact add (add (add (add (add (add hn hd) hk) ht) (mul (mul (const 1000 _) hk) hs)) he) hb |
| lemma bound_sourceRequest : Bound (OnlineSample.requestMeasure ∘ sourceRequest) requestSize := by |
| have hp := NumericBounds.comp bound_plan bound_precision |
| have hw := NumericBounds.weight (work_of_polynomial polynomial_sourceRequest X |
| (fun x=>by simpa using weight_le_requestSize x)) |
| apply NumericBounds.mono (NumericBounds.add hp (NumericBounds.mul (NumericBounds.const 4 _) hw)) |
| intro x |
| dsimp only [Function.comp_apply,sourceRequest,OnlineSample.requestMeasure,OnlineSample.scansMeasure, |
| OnlineSample.pairMeasure,OnlineSample.marksMeasure] |
| have h1:=weight_fst (sourceRequest x) |
| have h2:=weight_fst (sourceRequest x).1 |
| have h3:=weight_snd (sourceRequest x) |
| dsimp only [sourceRequest] at h1 h2 h3 |
| omega |
| end MatchingFPRAS.OnlineSchedule |
|
|
| end OAI |
|
|
