| 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] |
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| 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 |
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| end OAI |
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