| 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 |
|
|
