| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Machines.TreeFiniteOps |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.TreeTyped |
| variable {A B : Type} [Codec A] [Codec B] [LinearOrder A] [LinearOrder B] |
|
|
| def boolOr : Realizer (fun x : Bool×Bool=>x.1 || x.2) := |
| (choose first (constant true) second).congr (by intro x; cases x.1 <;> rfl) |
| lemma polynomial_boolOr : PolynomialTime boolOr := by |
| unfold boolOr |
| exact polynomial_congr _ (polynomial_choose polynomial_first (polynomial_constant true) polynomial_second) |
|
|
| def colexLe (E : Realizer (fun x : A×A=>decide (x.1=x.2))) |
| (R : Realizer (fun x : A×A=>decide (x.1≤x.2))) : |
| Realizer (fun x : Finset A×Finset A=>decide (toColex x.1≤toColex x.2)) := |
| (composition (pair (composition (pair second first) (finsetSDiff E)) (finsetSDiff E)) |
| (finsetAll (composition (pair second first) (finsetAny R)))).congr (by |
| intro x |
| apply Bool.eq_iff_iff.mpr |
| simp only [Function.comp_apply,decide_eq_true_eq,Finset.mem_sdiff] |
| rw [Finset.Colex.toColex_le_toColex] |
| constructor |
| · intro h a ha hn |
| obtain ⟨b,⟨hb,hbn⟩,hab⟩:=h a ⟨ha,hn⟩ |
| exact ⟨b,hb,hbn,hab⟩ |
| · intro h a ⟨ha,hn⟩ |
| obtain ⟨b,hb,hbn,hab⟩:=h ha hn |
| exact ⟨b,⟨hb,hbn⟩,hab⟩) |
| lemma polynomial_colexLe {E : Realizer (fun x : A×A=>decide (x.1=x.2))} |
| {R : Realizer (fun x : A×A=>decide (x.1≤x.2))} |
| (hE : PolynomialTime E) (hR : PolynomialTime R) : PolynomialTime (colexLe E R) := by |
| unfold colexLe |
| apply polynomial_congr |
| exact polynomial_composition (polynomial_pair |
| (polynomial_composition (polynomial_pair polynomial_second polynomial_first) (polynomial_finsetSDiff hE)) |
| (polynomial_finsetSDiff hE)) |
| (polynomial_finsetAll (polynomial_composition (polynomial_pair polynomial_second polynomial_first) |
| (polynomial_finsetAny hR))) |
|
|
| def lexLe (E : Realizer (fun x : A×A=>decide (x.1=x.2))) |
| (R : Realizer (fun x : A×A=>decide (x.1≤x.2))) |
| (S : Realizer (fun x : B×B=>decide (x.1≤x.2))) : |
| Realizer (fun x : (A×B)×(A×B)=>decide (toLex x.1≤toLex x.2)) := |
| (composition (pair (composition (pair first.fst second.fst) R) |
| (composition (pair (composition (composition (pair first.fst second.fst) E) boolNot) |
| (composition (pair first.snd second.snd) S)) boolOr)) boolAnd).congr (by |
| intro x |
| apply Bool.eq_iff_iff.mpr |
| simp only [Function.comp_apply,Bool.and_eq_true,Bool.or_eq_true,Bool.not_eq_true', |
| decide_eq_true_eq,decide_eq_false_iff_not,Prod.Lex.le_iff] |
| rcases x with ⟨⟨a,b⟩,⟨c,d⟩⟩ |
| dsimp only |
| constructor |
| · rintro ⟨h,h'⟩ |
| rcases lt_or_eq_of_le h with h | h |
| · exact Or.inl h |
| · exact Or.inr ⟨h,h'.resolve_left (not_not.mpr h)⟩ |
| · rintro (h|⟨h,h'⟩) |
| · exact ⟨h.le,Or.inl (ne_of_lt h)⟩ |
| · exact ⟨h.le,Or.inr h'⟩) |
| lemma polynomial_lexLe {E : Realizer (fun x : A×A=>decide (x.1=x.2))} |
| {R : Realizer (fun x : A×A=>decide (x.1≤x.2))} |
| {S : Realizer (fun x : B×B=>decide (x.1≤x.2))} |
| (hE : PolynomialTime E) (hR : PolynomialTime R) (hS : PolynomialTime S) : |
| PolynomialTime (lexLe E R S) := by |
| unfold lexLe |
| exact polynomial_congr _ (polynomial_composition (polynomial_pair |
| (polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) (polynomial_fst polynomial_second)) hR) |
| (polynomial_composition (polynomial_pair |
| (polynomial_composition (polynomial_composition |
| (polynomial_pair (polynomial_fst polynomial_first) (polynomial_fst polynomial_second)) hE) polynomial_boolNot) |
| (polynomial_composition (polynomial_pair (polynomial_snd polynomial_first) (polynomial_snd polynomial_second)) hS)) polynomial_boolOr)) |
| polynomial_boolAnd) |
| end MatchingFPRAS.TreeTyped |
| |
| end OAI |
| |
