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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 | 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
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