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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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import Mathlib
import OAI.Combinatorics.MatchingCount.Complexity.TreeNumericCosts
import OAI.Combinatorics.MatchingCount.Complexity.TreePolynomialGrid
import OAI.Combinatorics.MatchingCount.Machines.OnlineFrame
import OAI.Combinatorics.MatchingCount.Complexity.OnlineCost
namespace OAI
namespace MatchingFPRAS.PhysicalStream.Experiment
variable {A : Type}
lemma worst_affine (p : Experiment A) (g : A→ℕ) (k c : ℕ) :
p.worst (fun a=>k*g a+c)=k*p.worst g+c := by
induction p with
| done a => rfl
| read p ih =>
simp only [worst,ih]
rcases le_total ((p false).worst g) ((p true).worst g) with h|h
· rw [max_eq_right h,max_eq_right (Nat.add_le_add_right (Nat.mul_le_mul_left k h) c)]
· rw [max_eq_left h,max_eq_left (Nat.add_le_add_right (Nat.mul_le_mul_left k h) c)]
| delay n p ih => exact ih
lemma worst_const_add (p : Experiment A) (c : ℕ) (g : A→ℕ) :
p.worst (fun a=>c+g a)=c+p.worst g := by
simpa only [one_mul,Nat.add_comm c] using worst_affine p g 1 c
lemma worst_add_const (p : Experiment A) (g : A→ℕ) (c : ℕ) :
p.worst (fun a=>g a+c)=p.worst g+c := by
simpa only [one_mul] using worst_affine p g 1 c
lemma worst_add_le (p : Experiment A) (f g : A→ℕ) :
p.worst (fun a=>f a+g a)≤p.worst f+p.worst g := by
exact worst_le _ _ _ ((all_worst p f).and (all_worst p g) |>.mono (fun _ h=>Nat.add_le_add h.1 h.2))
end MatchingFPRAS.PhysicalStream.Experiment
namespace MatchingFPRAS.OnlineTree
open TreeTyped PhysicalStream Polynomial
variable {A B C : Type} [Codec A] [Codec B] [Codec C]
lemma mapSecond_cost {f : B→Experiment C} (F : Realizer f) (x : A×B) :
(mapSecond F).cost x=F.cost x.2+8*weight x.1+22 := rfl
omit [Codec B] in
lemma mapSecond_weight [Codec B] {f : B→Experiment C} (x : A×B) :
((f x.2).map (Prod.mk x.1)).worst weight=1+weight x.1+(f x.2).worst weight := by
rw [Experiment.worst_map]
simp only [Function.comp_def,weight_prod]
exact Experiment.worst_const_add _ _ _
lemma work_mapSecond {f : B→Experiment C} {F : Realizer f} {m : B→ℕ} {n : A×B→ℕ}
(hF : WorkBound F m) (p q : Polynomial ℕ)
(hm : ∀ x,m x.2≤p.eval (n x)) (hw : ∀ x,weight x.1≤q.eval (n x)) :
WorkBound (mapSecond F) n := by
obtain ⟨r,hr⟩:=hF
refine ⟨r.comp p+9*q+23,fun x=>?_⟩
change (mapSecond F).cost x+((f x.2).map (Prod.mk x.1)).worst weight≤_
rw [mapSecond_cost,mapSecond_weight]
have h:=(hr x.2).trans (polynomial_eval_monotone r (hm x))
dsimp only at h
have hh:=hw x
simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_comp,Polynomial.eval_ofNat]
omega
lemma polynomial_mapSecond {f : B→Experiment C} {F : Realizer f} (hF : PolynomialTime F) :
PolynomialTime (mapSecond (A:=A) F) :=
work_mapSecond hF X X (fun x=>by simpa using weight_snd x) (fun x=>by simpa using weight_fst x)
lemma pair_cost {f : A→Experiment B} {g : A→Experiment C} (F : Realizer f) (G : Realizer g) (a : A) :
(pair F G).cost a=F.cost a+G.cost a+45*weight a+26*(f a).worst weight+137 := by
simp only [pair,Realizer.congr,composition,deterministic,mapSecond,TreeTyped.pair,
TreeTyped.identity,TreeTyped.first,TreeTyped.second,Experiment.worst_map,
Experiment.worst,Function.comp_def,id_eq]
change 1+1+11*weight a+6*weight a+31+
(F.cost a+8*weight a+22+(f a).worst (fun b=>
3*weight a+7+(6*weight a+weight b+11)+11*weight (a,b)+6*weight b+31+
(G.cost a+8*weight b+22)))=_
simp only [weight_prod]
have h : (fun b : B=>3*weight a+7+(6*weight a+weight b+11)+11*(1+weight a+weight b)+6*weight b+31+
(G.cost a+8*weight b+22))=(fun b=>26*weight b+(20*weight a+G.cost a+82)) := by funext b; omega
rw [h,Experiment.worst_affine]
omega
omit [Codec A] in
lemma pair_weight [Codec A] {f : A→Experiment B} {g : A→Experiment C} (a : A) :
((f a).bind (fun b=>(g a).map (Prod.mk b))).worst weight=
1+(f a).worst weight+(g a).worst weight := by
rw [Experiment.worst_bind]
simp only [Experiment.worst_map,Function.comp_def,weight_prod,
Experiment.worst_const_add,Experiment.worst_add_const]
lemma work_pair {f : A→Experiment B} {g : A→Experiment C} {F : Realizer f} {G : Realizer g}
{m : A→ℕ} (hF : WorkBound F m) (hG : WorkBound G m)
(p : Polynomial ℕ) (hw : ∀ a,weight a≤p.eval (m a)) : WorkBound (pair F G) m := by
obtain ⟨q,hq⟩:=hF
obtain ⟨r,hr⟩:=hG
refine ⟨27*q+r+45*p+138,fun a=>?_⟩
change (pair F G).cost a+((f a).bind (fun b=>(g a).map (Prod.mk b))).worst weight≤_
rw [pair_cost,pair_weight]
have h1:=hq a
have h2:=hr a
have h3:=hw a
simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat]
omega
lemma polynomial_pair {f : A→Experiment B} {g : A→Experiment C} {F : Realizer f} {G : Realizer g}
(hF : PolynomialTime F) (hG : PolynomialTime G) : PolynomialTime (pair F G) :=
work_pair hF hG X (by intro a; simp)
lemma work_choose {f g : A→Experiment B} {F : Realizer f} {G : Realizer g} {m : A→ℕ}
(hF : WorkBound F m) (hG : WorkBound G m) : WorkBound (choose F G) m := by
obtain ⟨p,hp⟩:=hF
obtain ⟨q,hq⟩:=hG
refine ⟨p+q+1,fun a=>?_⟩
change 1+max (F.cost a) (G.cost a)+max ((f a).worst weight) ((g a).worst weight)≤_
have h1:=hp a
have h2:=hq a
simp only [Polynomial.eval_add,Polynomial.eval_one]
omega
lemma polynomial_choose {f g : A→Experiment B} {F : Realizer f} {G : Realizer g}
(hF : PolynomialTime F) (hG : PolynomialTime G) : PolynomialTime (choose F G) := work_choose hF hG
end MatchingFPRAS.OnlineTree
end OAI
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