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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import Mathlib
import OAI.Combinatorics.MatchingCount.Arithmetic.TreeFraction
import OAI.Combinatorics.MatchingCount.Complexity.TreePolynomialDivision
import OAI.Combinatorics.MatchingCount.Complexity.TreePolynomialLoop
namespace OAI
namespace MatchingFPRAS.TreeTyped
open Polynomial
lemma polynomial_boolNot : PolynomialTime boolNot := by
unfold boolNot
exact polynomial_congr _ (polynomial_choose polynomial_identity (polynomial_constant false) (polynomial_constant true))
lemma polynomial_boolAnd : PolynomialTime boolAnd := by
unfold boolAnd
exact polynomial_congr _ (polynomial_choose polynomial_first polynomial_second (polynomial_constant false))
lemma polynomial_natEqual : PolynomialTime natEqual := by
unfold natEqual
apply polynomial_congr
apply polynomial_choose polynomial_less (polynomial_constant false)
exact polynomial_choose (polynomial_composition (polynomial_pair polynomial_second polynomial_first) polynomial_less)
(polynomial_constant false) (polynomial_constant true)
lemma polynomial_quotient : PolynomialTime quotient := polynomial_composition polynomial_division polynomial_first
lemma polynomial_remainder : PolynomialTime remainder := polynomial_composition polynomial_division polynomial_second
lemma polynomial_digits : PolynomialTime digits := by
refine ⟨X+1,?_⟩
intro a
change 1+weight a≤_
simp only [Polynomial.eval_add,Polynomial.eval_X,Polynomial.eval_one]
omega
lemma polynomial_bitSize : PolynomialTime bitSize := by
unfold bitSize
exact polynomial_congr _ (polynomial_composition polynomial_digits polynomial_listLength)
lemma polynomial_sortedPair : PolynomialTime sortedPairRealizer := by
unfold sortedPairRealizer
exact polynomial_congr _ (polynomial_choose polynomial_less (polynomial_pair polynomial_second polynomial_first) polynomial_identity)
lemma polynomial_gcdStep : PolynomialTime gcdStepRealizer := by
unfold gcdStepRealizer
exact polynomial_congr _ (polynomial_choose
(polynomial_composition (polynomial_pair polynomial_second (polynomial_constant 0)) polynomial_natEqual)
polynomial_identity (polynomial_pair polynomial_second polynomial_remainder))
lemma gcdStep_size_le (x : ℕ×ℕ) : (gcdStep x).1.size+(gcdStep x).2.size≤x.1.size+x.2.size := by
dsimp only [gcdStep]
split_ifs with h
· exact le_rfl
· simp only
have hm:=Nat.size_le_size (Nat.mod_le x.1 x.2)
omega
lemma polynomial_gcd : PolynomialTime gcd := by
unfold gcd
apply polynomial_congr
apply polynomial_composition _ polynomial_first
apply polynomial_boundedRepeat
(polynomial_pair (polynomial_composition (polynomial_pair (polynomial_composition polynomial_first polynomial_bitSize)
(polynomial_composition polynomial_second polynomial_bitSize)) polynomial_add) polynomial_sortedPair)
(polynomial_composition polynomial_second polynomial_gcdStep) X (4*X+3)
(fun x _ z=>z.1.size+z.2.size≤x.1.size+x.2.size)
· intro x
simp only [Polynomial.eval_X,Function.comp_apply]
have h1:=size_le_weight x.1
have h2:=size_le_weight x.2
rw [←Prod.eta x,weight_prod]
try dsimp only
omega
· intro x
exact (sortedPair_size x).le
· intro x _iteration _bounded z hz
exact (gcdStep_size_le z).trans hz
· intro x _iteration z hz
have ha:=weight_nat_le z.1
have hb:=weight_nat_le z.2
have hx:=size_le_weight x.1
have hy:=size_le_weight x.2
rw [←Prod.eta z,weight_prod]
simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X]
rw [←Prod.eta x,weight_prod]
try dsimp only
omega
end MatchingFPRAS.TreeTyped
namespace MatchingFPRAS.TreeFraction
open TreeTyped
lemma polynomial_normalize : PolynomialTime normalizeRealizer := by
unfold normalizeRealizer
apply polynomial_congr
apply polynomial_choose
· exact polynomial_composition (polynomial_pair polynomial_second (polynomial_constant 0)) polynomial_natEqual
· exact polynomial_constant (0,1)
· exact polynomial_pair
(polynomial_composition (polynomial_pair polynomial_first polynomial_gcd) polynomial_quotient)
(polynomial_composition (polynomial_pair polynomial_second polynomial_gcd) polynomial_quotient)
lemma polynomial_mul : PolynomialTime mulRealizer := by
unfold mulRealizer
exact polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) (polynomial_fst polynomial_second)) polynomial_multiply)
(polynomial_composition (polynomial_pair (polynomial_snd polynomial_first) (polynomial_snd polynomial_second)) polynomial_multiply)) polynomial_normalize
lemma polynomial_add : PolynomialTime addRealizer := by
unfold addRealizer
apply polynomial_composition _ polynomial_normalize
apply polynomial_pair
· apply polynomial_composition _ TreeTyped.polynomial_add
exact polynomial_pair
(polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) (polynomial_snd polynomial_second)) polynomial_multiply)
(polynomial_composition (polynomial_pair (polynomial_fst polynomial_second) (polynomial_snd polynomial_first)) polynomial_multiply)
· exact polynomial_composition (polynomial_pair (polynomial_snd polynomial_first) (polynomial_snd polynomial_second)) polynomial_multiply
lemma polynomial_div : PolynomialTime divRealizer := by
unfold divRealizer
exact polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) (polynomial_snd polynomial_second)) polynomial_multiply)
(polynomial_composition (polynomial_pair (polynomial_snd polynomial_first) (polynomial_fst polynomial_second)) polynomial_multiply)) polynomial_normalize
lemma polynomial_less : PolynomialTime lessRealizer := by
unfold lessRealizer
exact polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) (polynomial_snd polynomial_second)) polynomial_multiply)
(polynomial_composition (polynomial_pair (polynomial_fst polynomial_second) (polynomial_snd polynomial_first)) polynomial_multiply)) TreeTyped.polynomial_less
end MatchingFPRAS.TreeFraction
end OAI
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