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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.TreeFinite
namespace OAI
namespace MatchingFPRAS.TreeTyped
open Polynomial
variable {A B C : Type} [Codec A] [Codec B] [Codec C]
def all : Realizer (fun bs : List Bool=>bs.all id) :=
(composition (composition (map boolNot) any) boolNot).congr (by
intro bs
change (!((bs.map Bool.not).any id))=bs.all id
induction bs with
| nil => rfl
| cons b bs ih => cases b <;> simp_all)
lemma polynomial_all : PolynomialTime all := by
unfold all
exact polynomial_congr _ (polynomial_composition
(polynomial_composition (polynomial_map polynomial_boolNot) polynomial_any) polynomial_boolNot)
def pairEqual [DecidableEq A] [DecidableEq B]
(E : Realizer (fun x : A×A=>decide (x.1=x.2)))
(F : Realizer (fun x : B×B=>decide (x.1=x.2))) :
Realizer (fun x : (A×B)×(A×B)=>decide (x.1=x.2)) :=
(composition (pair (composition (pair first.fst second.fst) E)
(composition (pair first.snd second.snd) F)) boolAnd).congr (by
intro x
apply Bool.eq_iff_iff.mpr
simp only [Function.comp_apply,Bool.and_eq_true,decide_eq_true_eq,Prod.ext_iff])
lemma polynomial_pairEqual [DecidableEq A] [DecidableEq B]
{E : Realizer (fun x : A×A=>decide (x.1=x.2))}
{F : Realizer (fun x : B×B=>decide (x.1=x.2))}
(hE : PolynomialTime E) (hF : PolynomialTime F) : PolynomialTime (pairEqual E F) := by
unfold pairEqual
exact polynomial_congr _ (polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) (polynomial_fst polynomial_second)) hE)
(polynomial_composition (polynomial_pair (polynomial_snd polynomial_first) (polynomial_snd polynomial_second)) hF)) polynomial_boolAnd)
def leNat : Realizer (fun x : ℕ×ℕ=>decide (x.1≤x.2)) :=
(composition (composition (pair second first) less) boolNot).congr (by
intro x
apply Bool.eq_iff_iff.mpr
simp only [Function.comp_apply,Bool.not_eq_true',decide_eq_false_iff_not,decide_eq_true_eq,not_lt])
lemma polynomial_leNat : PolynomialTime leNat := by
unfold leNat
exact polynomial_congr _ (polynomial_composition
(polynomial_composition (polynomial_pair polynomial_second polynomial_first) polynomial_less) polynomial_boolNot)
variable [LinearOrder A]
def finsetAny {p : B×A → Bool} (P : Realizer p) :
Realizer (fun x : B×Finset A=>decide (∃ a∈x.2,p (x.1,a)=true)) :=
(composition (composition (pair first (composition second finsetList)) (mapWith P)) any).congr (by
intro x
apply Bool.eq_iff_iff.mpr
simp only [Function.comp_apply,List.any_map,List.any_eq_true,id_eq,decide_eq_true_eq,Finset.mem_sort])
lemma polynomial_finsetAny {p : B×A → Bool} {P : Realizer p} (hP : PolynomialTime P) :
PolynomialTime (finsetAny P) := by
unfold finsetAny
exact polynomial_congr _ (polynomial_composition (polynomial_composition
(polynomial_pair polynomial_first (polynomial_composition polynomial_second polynomial_finsetList))
(polynomial_mapWith hP)) polynomial_any)
def finsetAll {p : B×A → Bool} (P : Realizer p) :
Realizer (fun x : B×Finset A=>decide (∀ a∈x.2,p (x.1,a)=true)) :=
(composition (composition (pair first (composition second finsetList)) (mapWith P)) all).congr (by
intro x
apply Bool.eq_iff_iff.mpr
simp only [Function.comp_apply,List.all_map,List.all_eq_true,id_eq,decide_eq_true_eq,Finset.mem_sort])
lemma polynomial_finsetAll {p : B×A → Bool} {P : Realizer p} (hP : PolynomialTime P) :
PolynomialTime (finsetAll P) := by
unfold finsetAll
exact polynomial_congr _ (polynomial_composition (polynomial_composition
(polynomial_pair polynomial_first (polynomial_composition polynomial_second polynomial_finsetList))
(polynomial_mapWith hP)) polynomial_all)
def finsetSubset (E : Realizer (fun x : A×A=>decide (x.1=x.2))) :
Realizer (fun x : Finset A×Finset A=>decide (x.1⊆x.2)) :=
(composition (pair second first) (finsetAll (composition (pair second first) (finsetMember E)))).congr (by
intro x; simp only [Function.comp_apply,decide_eq_true_eq]; rfl)
lemma polynomial_finsetSubset {E : Realizer (fun x : A×A=>decide (x.1=x.2))}
(hE : PolynomialTime E) : PolynomialTime (finsetSubset E) := by
unfold finsetSubset
exact polynomial_congr _ (polynomial_composition (polynomial_pair polynomial_second polynomial_first)
(polynomial_finsetAll (polynomial_composition (polynomial_pair polynomial_second polynomial_first) (polynomial_finsetMember hE))))
def finsetEqual (E : Realizer (fun x : A×A=>decide (x.1=x.2))) :
Realizer (fun x : Finset A×Finset A=>decide (x.1=x.2)) :=
(composition (pair (finsetSubset E) (composition (pair second first) (finsetSubset E))) boolAnd).congr (by
intro x
apply Bool.eq_iff_iff.mpr
simp only [Function.comp_apply,Bool.and_eq_true,decide_eq_true_eq,Finset.Subset.antisymm_iff])
lemma polynomial_finsetEqual {E : Realizer (fun x : A×A=>decide (x.1=x.2))}
(hE : PolynomialTime E) : PolynomialTime (finsetEqual E) := by
unfold finsetEqual
exact polynomial_congr _ (polynomial_composition (polynomial_pair (polynomial_finsetSubset hE)
(polynomial_composition (polynomial_pair polynomial_second polynomial_first) (polynomial_finsetSubset hE))) polynomial_boolAnd)
def finsetSDiff (E : Realizer (fun x : A×A=>decide (x.1=x.2))) :
Realizer (fun x : Finset A×Finset A=>x.1\x.2) :=
(composition (pair second first) (finsetFilterWith (composition
(composition (pair second first) (finsetMember E)) boolNot))).congr (by intro x; ext a; simp)
lemma polynomial_finsetSDiff {E : Realizer (fun x : A×A=>decide (x.1=x.2))}
(hE : PolynomialTime E) : PolynomialTime (finsetSDiff E) := by
unfold finsetSDiff
exact polynomial_congr _ (polynomial_composition (polynomial_pair polynomial_second polynomial_first)
(polynomial_finsetFilterWith (polynomial_composition
(polynomial_composition (polynomial_pair polynomial_second polynomial_first) (polynomial_finsetMember hE)) polynomial_boolNot)))
def finsetInter (E : Realizer (fun x : A×A=>decide (x.1=x.2))) :
Realizer (fun x : Finset A×Finset A=>x.1∩x.2) :=
(composition (pair second first) (finsetFilterWith (composition
(pair second first) (finsetMember E)))).congr (by
intro x; ext a
simp only [Function.comp_apply,Finset.mem_filter,decide_eq_true_eq,Finset.mem_inter])
lemma polynomial_finsetInter {E : Realizer (fun x : A×A=>decide (x.1=x.2))}
(hE : PolynomialTime E) : PolynomialTime (finsetInter E) := by
unfold finsetInter
exact polynomial_congr _ (polynomial_composition (polynomial_pair polynomial_second polynomial_first)
(polynomial_finsetFilterWith (polynomial_composition
(polynomial_pair polynomial_second polynomial_first) (polynomial_finsetMember hE))))
def finsetErase (E : Realizer (fun x : A×A=>decide (x.1=x.2))) :
Realizer (fun x : A×Finset A=>x.2.erase x.1) :=
(finsetFilterWith (composition E boolNot)).congr (by
intro x; ext a
simp only [Function.comp_apply,Finset.mem_filter,Bool.not_eq_true',decide_eq_false_iff_not,Finset.mem_erase]
tauto)
lemma polynomial_finsetErase {E : Realizer (fun x : A×A=>decide (x.1=x.2))}
(hE : PolynomialTime E) : PolynomialTime (finsetErase E) := by
unfold finsetErase
exact polynomial_congr _ (polynomial_finsetFilterWith (polynomial_composition hE polynomial_boolNot))
end MatchingFPRAS.TreeTyped
end OAI
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