| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Machines.OnlineFold |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineTree |
| open TreeTyped PhysicalStream Polynomial |
|
|
| variable {A B : Type} [Codec A] [Codec B] |
|
|
| omit [Codec B] in |
| lemma fold_cost_bound (f : A×B→Experiment B) (t : A×B→ℕ) |
| (as : List A) (b : B) (P : ℕ→B→Prop) (C M : ℕ) |
| (hb : P as.length b) (hw : weight as≤M) |
| (hs : ∀ a∈as,∀ k<as.length,∀ b,P (k+1) b→(f (a,b)).All (P k)) |
| (hc : ∀ a∈as,∀ k<as.length,∀ b,P (k+1) b→t (a,b)≤C) : |
| Fold.cost f t as b≤as.length*(C+14*M+34)+10 ∧ (foldM f as b).All (P 0) := by |
| induction as generalizing b with |
| | nil => exact ⟨by simp [Fold.cost],hb⟩ |
| | cons a as ih => |
| have hwa : weight a≤M := by rw [weight_cons] at hw; omega |
| have hwt : weight as≤M := by rw [weight_cons] at hw; omega |
| have hb' : P (as.length+1) b := hb |
| have hstep := hs a (by simp) as.length (by simp) b hb' |
| have htime := hc a (by simp) as.length (by simp) b hb' |
| have htt : (f (a,b)).All (fun v=>Fold.cost f t as v≤as.length*(C+14*M+34)+10 ∧ |
| (foldM f as v).All (P 0)) := hstep.mono (fun v hv=>ih v hv hwt |
| (fun a ha k hk v hv=>hs a (by simp [ha]) k (by simp only [List.length_cons]; omega) v hv) |
| (fun a ha k hk v hv=>hc a (by simp [ha]) k (by simp only [List.length_cons]; omega) v hv)) |
| constructor |
| · have hworst := Experiment.worst_le _ _ _ (htt.mono (fun _ h=>h.1)) |
| change t (a,b)+6*weight a+8*weight as+34+_≤_ |
| simp only [List.length_cons,Nat.add_mul,Nat.one_mul] |
| omega |
| · rw [foldM,Experiment.all_bind] |
| exact htt.mono (fun _ h=>h.2) |
|
|
| omit [Codec A] in |
| lemma all_fold_weight (f : A×B→Experiment B) (as : List A) (b : B) (C : ℕ) |
| (hs : ∀ a∈as,∀ b,(f (a,b)).All (fun v=>weight v≤weight b+C)) : |
| (foldM f as b).All (fun v=>weight v≤weight b+as.length*C) := by |
| induction as generalizing b with |
| | nil => simp [foldM,Experiment.All] |
| | cons a as ih => |
| rw [foldM,Experiment.all_bind] |
| exact (hs a (by simp) b).mono (fun c hc=> |
| (ih c (fun a ha b=>hs a (by simp [ha]) b)).mono (fun v hv=>by |
| simp only [List.length_cons,Nat.add_mul,Nat.one_mul]; omega)) |
| end MatchingFPRAS.OnlineTree |
|
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| end OAI |
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