| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Complexity.TreeCostMeasure |
| import OAI.Combinatorics.MatchingCount.Machines.OnlineTree |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.PhysicalStream.Experiment |
| variable {A B : Type} |
| lemma worst_mono (p : Experiment A) (f g : A→ℕ) (h : p.All (fun a=>f a≤g a)) : |
| p.worst f≤p.worst g := by |
| induction p with |
| | done a => exact h |
| | read p ih => exact max_le_max (ih false h.1) (ih true h.2) |
| | delay n p ih => exact ih h |
| lemma All.and {p : Experiment A} {P Q : A→Prop} (hp : p.All P) (hq : p.All Q) : |
| p.All (fun a=>P a ∧ Q a) := by |
| induction p with |
| | done a => exact ⟨hp,hq⟩ |
| | read p ih => exact ⟨ih false hp.1 hq.1,ih true hp.2 hq.2⟩ |
| | delay n p ih => exact ih hp hq |
| lemma worst_const (p : Experiment A) (n : ℕ) : p.worst (fun _=>n)=n := by |
| induction p with |
| | done a => rfl |
| | read p ih => simp only [worst,ih, max_self] |
| | delay n p ih => exact ih |
| lemma worst_bind (p : Experiment A) (f : A→Experiment B) (h : B→ℕ) : |
| (p.bind f).worst h=p.worst (fun a=>(f a).worst h) := by |
| induction p with |
| | done a => rfl |
| | read p ih => simp only [bind,worst,ih] |
| | delay n p ih => exact ih |
| lemma worst_map (p : Experiment A) (f : A→B) (h : B→ℕ) : |
| (p.map f).worst h=p.worst (h ∘ f) := worst_bind _ _ _ |
| lemma worst_eq_of_mean_eq {p q : Experiment A} (h : ∀ f : A→ℚ,p.mean f=q.mean f) (g : A→ℕ) : |
| p.worst g=q.worst g := by |
| apply Nat.le_antisymm |
| · exact worst_le _ _ _ (all_of_mean_eq h _ (all_worst q g)) |
| · exact worst_le _ _ _ (all_of_mean_eq (fun f=>(h f).symm) _ (all_worst p g)) |
| end MatchingFPRAS.PhysicalStream.Experiment |
|
|
| namespace MatchingFPRAS.OnlineTree |
| open TreeTyped PhysicalStream Polynomial |
|
|
| variable {A B C : Type} [Codec A] [Codec B] [Codec C] |
| def WorkBound {f : A→Experiment B} (F : Realizer f) (measure : A→ℕ) : Prop := |
| ∃ p : Polynomial ℕ,∀ a,F.cost a+(f a).worst weight≤p.eval (measure a) |
| abbrev PolynomialTime {f : A→Experiment B} (F : Realizer f) : Prop := WorkBound F weight |
| lemma work_deterministic {f : A→B} {F : TreeTyped.Realizer f} {m : A→ℕ} |
| (hF : TreeTyped.WorkBound F m) : WorkBound (deterministic F) m := hF |
| lemma work_congr {f g : A→Experiment B} {F : Realizer f} {m : A→ℕ} |
| (h : ∀ a w,(f a).mean w=(g a).mean w) (hF : WorkBound F m) : WorkBound (F.congr h) m := by |
| obtain ⟨p,hp⟩:=hF |
| refine ⟨p,fun a=>?_⟩ |
| change F.cost a+(g a).worst weight≤_ |
| rw [←Experiment.worst_eq_of_mean_eq (h a)] |
| exact hp a |
| lemma work_remeasure {f : A→Experiment B} {F : Realizer f} {m n : A→ℕ} |
| (hF : WorkBound F m) (p : Polynomial ℕ) (hm : ∀ a,m a≤p.eval (n a)) : WorkBound F n := by |
| obtain ⟨q,hq⟩:=hF |
| exact ⟨q.comp p,fun a=>(hq a).trans (by simpa using polynomial_eval_monotone q (hm a))⟩ |
| lemma work_of_polynomial {f : A→Experiment B} {F : Realizer f} {m : A→ℕ} |
| (hF : PolynomialTime F) (p : Polynomial ℕ) (hm : ∀ a,weight a≤p.eval (m a)) : WorkBound F m := |
| work_remeasure hF p hm |
| lemma work_composition {f : A→Experiment B} {g : B→Experiment C} {F : Realizer f} {G : Realizer g} |
| {m : A→ℕ} {n : B→ℕ} (hF : WorkBound F m) (hG : WorkBound G n) |
| (p : Polynomial ℕ) (hm : ∀ a,(f a).All (fun b=>n b≤p.eval (m a))) : |
| WorkBound (composition F G) m := by |
| obtain ⟨q,hq⟩:=hF |
| obtain ⟨r,hr⟩:=hG |
| refine ⟨q+2*(r.comp p),fun a=>?_⟩ |
| have hb : (f a).All (fun b=>G.cost b+(g b).worst weight≤r.eval (p.eval (m a))) := |
| (hm a).mono (fun b hb=>(hr b).trans (polynomial_eval_monotone r hb)) |
| have h1:=Experiment.worst_le _ _ _ (hb.mono (fun b hb=>Nat.le_trans (Nat.le_add_right _ _) hb)) |
| have h2:=Experiment.worst_le _ _ _ (hb.mono (fun b hb=>Nat.le_trans (Nat.le_add_left _ _) hb)) |
| have h3:=hq a |
| change F.cost a+(f a).worst G.cost+((f a).bind g).worst weight≤_ |
| rw [Experiment.worst_bind] |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_comp] |
| omega |
| lemma work_then_polynomial {f : A→Experiment B} {g : B→Experiment C} {F : Realizer f} {G : Realizer g} |
| {m : A→ℕ} (hF : WorkBound F m) (hG : PolynomialTime G) : WorkBound (composition F G) m := by |
| obtain ⟨p,hp⟩:=hF |
| apply work_composition ⟨p,hp⟩ hG p |
| intro a |
| exact (Experiment.all_worst (f a) weight).mono (fun b hb=>hb.trans (Nat.le_trans (Nat.le_add_left _ _) (hp a))) |
| lemma polynomial_composition {f : A→Experiment B} {g : B→Experiment C} {F : Realizer f} {G : Realizer g} |
| (hF : PolynomialTime F) (hG : PolynomialTime G) : PolynomialTime (composition F G) := work_then_polynomial hF hG |
| lemma polynomial_deterministic {f : A→B} {F : TreeTyped.Realizer f} (hF : TreeTyped.PolynomialTime F) : |
| PolynomialTime (deterministic F) := hF |
| end MatchingFPRAS.OnlineTree |
|
|
| end OAI |
|
|
