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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Complexity.TreeCostMeasure
namespace OAI
namespace MatchingFPRAS.TreeTyped
open Polynomial
variable {A B C : Type} [Codec A] [Codec B] [Codec C]
lemma work_pair {f : A → B} {g : A → C} {F : Realizer f} {G : Realizer g}
{measure : A → ℕ} (hF : WorkBound F measure) (hG : WorkBound G measure)
(u : Polynomial ℕ) (hu : ∀ a,weight a≤u.eval (measure a)) : WorkBound (pair F G) measure := by
obtain ⟨p,hp⟩:=hF
obtain ⟨q,hq⟩:=hG
refine ⟨7*p+q+11*u+32,fun a=>?_⟩
have h1:=hp a
have h2:=hq a
have h3:=hu a
simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat]
rw [weight_prod]
change F.cost a+G.cost a+11*weight a+6*weight (f a)+31+(1+weight (f a)+weight (g a))≤_
omega
omit [Codec C] in
lemma work_choose {f : A → Bool} {g h : A → B} {F : Realizer f} {G : Realizer g} {H : Realizer h}
{measure : A → ℕ} (hF : WorkBound F measure) (hG : WorkBound G measure) (hH : WorkBound H measure)
(u : Polynomial ℕ) (hu : ∀ a,weight a≤u.eval (measure a)) : WorkBound (choose F G H) measure := by
obtain ⟨p,hp⟩:=hF
obtain ⟨q,hq⟩:=hG
obtain ⟨r,hr⟩:=hH
refine ⟨p+q+r+7*u+18,fun a=>?_⟩
have h1:=hp a
have h2:=hq a
have h3:=hr a
have h4:=hu a
simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat]
change F.cost a+(if f a then G.cost a else H.cost a)+7*weight a+18+
weight (if f a then g a else h a)≤_
cases f a <;> simp only [Bool.false_eq_true,↓reduceIte] <;> omega
omit [Codec A] [Codec B] [Codec C] in
lemma factorial_numeric_work : WorkBound factorial id := by
unfold factorial
apply work_congr
apply work_boundedRepeat (work_of_polynomial (polynomial_pair polynomial_identity (polynomial_constant 1))
(4*X+1) (fun n=>by simpa using weight_nat_numeric n)) polynomial_multiply X (4*X^2+5)
(fun n k z=>k≤n ∧ z.size≤1+(n-k)*n)
· intro n; simp
· intro n; simp
· intro n k hk z hz
change k<n at hk
have he : n-k=(n-(k+1))+1:=by omega
have hh:=nat_size_mul (k+1) z
have hb : (k+1).size≤n := (Nat.size_le.mpr Nat.lt_two_pow_self).trans (by omega)
change k≤n ∧ ((k+1)*z).size≤1+(n-k)*n
refine ⟨by omega,?_⟩
have hbound : 1+(n-k)*n=1+(n-(k+1))*n+n := by rw [he]; ring
rw [hbound]
omega
· intro n k z hz
have hh:=weight_nat_le z
have hm:=Nat.mul_le_mul_right n (Nat.sub_le n k)
simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X,Polynomial.eval_pow,id_eq]
nlinarith
omit [Codec A] [Codec B] [Codec C] in
lemma pair_nat_weight_numeric (x : ℕ×ℕ) : weight x≤4*(x.1+x.2)+3 := by
have h1:=weight_nat_numeric x.1
have h2:=weight_nat_numeric x.2
rw [←Prod.eta x,weight_prod]
omega
omit [Codec A] [Codec B] [Codec C] in
lemma natChoose_numeric_work : WorkBound natChoose (fun x=>x.1+x.2) := by
have ht : ∀ {f : ℕ×ℕ → ℕ} {F : Realizer f},PolynomialTime F → WorkBound F (fun x=>x.1+x.2) :=
fun h=>work_of_polynomial h (4*X+3) (fun x=>by simpa using pair_nat_weight_numeric x)
have hb : ∀ {f : ℕ×ℕ → Bool} {F : Realizer f},PolynomialTime F → WorkBound F (fun x=>x.1+x.2) :=
fun h=>work_of_polynomial h (4*X+3) (fun x=>by simpa using pair_nat_weight_numeric x)
have hpair : ∀ {f g : ℕ×ℕ → ℕ} {F : Realizer f} {G : Realizer g},
WorkBound F (fun x=>x.1+x.2) → WorkBound G (fun x=>x.1+x.2) → WorkBound (pair F G) (fun x=>x.1+x.2) :=
fun h1 h2=>work_pair h1 h2 (4*X+3) (fun x=>by simpa using pair_nat_weight_numeric x)
have hf : WorkBound (composition (first (A:=ℕ) (B:=ℕ)) factorial) (fun x=>x.1+x.2) :=
work_composition (ht polynomial_first) factorial_numeric_work X (fun x=>by simp only [Polynomial.eval_X,id_eq]; omega)
have hs : WorkBound (composition (second (A:=ℕ) (B:=ℕ)) factorial) (fun x=>x.1+x.2) :=
work_composition (ht polynomial_second) factorial_numeric_work X (fun x=>by simp only [Polynomial.eval_X,id_eq]; omega)
have hd : WorkBound (composition subtract factorial) (fun x=>x.1+x.2) :=
work_composition (ht polynomial_subtract) factorial_numeric_work X (fun x=>by simp only [Polynomial.eval_X,id_eq]; omega)
unfold natChoose
apply work_congr
exact work_choose
(hb (polynomial_composition (polynomial_pair polynomial_second polynomial_first) polynomial_less))
(work_then_polynomial (hpair hf (work_then_polynomial (hpair hs hd) polynomial_multiply)) polynomial_quotient)
(work_choose (hb (polynomial_composition (polynomial_pair polynomial_first polynomial_second) polynomial_less))
(ht (polynomial_constant 0)) (ht (polynomial_constant 1)) (4*X+3) (fun x=>by simpa using pair_nat_weight_numeric x))
(4*X+3) (fun x=>by simpa using pair_nat_weight_numeric x)
omit [Codec B] [Codec C] in
lemma polynomial_boundedFactorial {f : A → ℕ} {F : Realizer f} (hF : PolynomialTime F)
(p : Polynomial ℕ) (hn : ∀ a,f a≤p.eval (weight a)) : PolynomialTime (composition F factorial) :=
work_composition hF factorial_numeric_work p hn
omit [Codec B] [Codec C] in
lemma polynomial_boundedChoose {f : A → ℕ×ℕ} {F : Realizer f} (hF : PolynomialTime F)
(p : Polynomial ℕ) (hn : ∀ a,(f a).1+(f a).2≤p.eval (weight a)) : PolynomialTime (composition F natChoose) :=
work_composition hF natChoose_numeric_work p hn
end MatchingFPRAS.TreeTyped
end OAI
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