| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Sampling.TreePairStep |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.ExecutablePair |
| open TreeTyped TreeFraction TreeTyped.SparseProgram |
|
|
| abbrev Context := List (ℕ×ℕ)×(List Fraction×List Fraction) |
| def endpoints (x : Context×ℕ) : ℕ×ℕ := x.1.1.getD x.2 (0,0) |
| def endpointsRealizer : Realizer endpoints := composition (pair first.fst second) (getDFast (0,0)) |
| lemma polynomial_endpoints : PolynomialTime endpointsRealizer := |
| polynomial_composition (polynomial_pair (polynomial_fst polynomial_first) polynomial_second) |
| (polynomial_getDFast (0,0)) |
|
|
| def incident (x : (ℕ×ℕ)×ℕ) : Bool := decide (x.1.1=x.2 ∨ x.1.2=x.2) |
| def incidentRealizer : Realizer incident := (composition (pair |
| (composition (pair first.fst second) natEqual) (composition (pair first.snd second) natEqual)) |
| boolOr).congr (by intro x; simp [incident]) |
| lemma polynomial_incident : PolynomialTime incidentRealizer := by |
| unfold incidentRealizer Realizer.fst Realizer.snd |
| apply polynomial_congr |
| tree_cost |
|
|
| def same (x : (ℕ×ℕ)×(ℕ×ℕ)) : Bool := |
| decide ((x.1.1=x.2.1 ∧ x.1.2=x.2.2) ∨ (x.1.1=x.2.2 ∧ x.1.2=x.2.1)) |
| def sameRealizer : Realizer same := (composition (pair |
| (composition (pair (composition (pair first.fst second.fst) natEqual) |
| (composition (pair first.snd second.snd) natEqual)) boolAnd) |
| (composition (pair (composition (pair first.fst second.snd) natEqual) |
| (composition (pair first.snd second.fst) natEqual)) boolAnd)) boolOr).congr |
| (by intro x; simp [same]) |
| lemma polynomial_same : PolynomialTime sameRealizer := by |
| unfold sameRealizer Realizer.fst Realizer.snd |
| apply polynomial_congr |
| tree_cost |
|
|
| def adjacent (x : (ℕ×ℕ)×(ℕ×ℕ)) : Bool := |
| incident (x.1,x.2.1) || incident (x.1,x.2.2) |
| def adjacentRealizer : Realizer adjacent := composition (pair |
| (composition (pair first second.fst) incidentRealizer) |
| (composition (pair first second.snd) incidentRealizer)) boolOr |
| lemma polynomial_adjacent : PolynomialTime adjacentRealizer := by |
| unfold adjacentRealizer adjacent Realizer.fst Realizer.snd |
| have := polynomial_incident |
| tree_cost |
|
|
| def p (x : Context×(ℕ×ℕ)) : Bool := incident (endpoints (x.1,x.2.1),x.2.2) |
| def q (x : Context×(ℕ×ℕ)) : Bool := same (endpoints (x.1,x.2.1),endpoints (x.1,x.2.2)) |
| def r (x : Context×(ℕ×(ℕ×ℕ))) : Bool := same (endpoints (x.1,x.2.1),x.2.2) |
| def t (x : Context×(ℕ×ℕ)) : Bool := adjacent (endpoints (x.1,x.2.1),endpoints (x.1,x.2.2)) |
| def u (x : Context×ℕ) : Fraction := x.1.2.1.getD x.2 (0,1) |
| def v (x : Context×ℕ) : Fraction := x.1.2.2.getD x.2 (0,1) |
| def pRealizer : Realizer p := composition (pair (composition (pair first second.fst) endpointsRealizer) second.snd) incidentRealizer |
| def qRealizer : Realizer q := composition (pair (composition (pair first second.fst) endpointsRealizer) |
| (composition (pair first second.snd) endpointsRealizer)) sameRealizer |
| def rRealizer : Realizer r := composition (pair (composition (pair first second.fst) endpointsRealizer) second.snd) sameRealizer |
| def tRealizer : Realizer t := composition (pair (composition (pair first second.fst) endpointsRealizer) |
| (composition (pair first second.snd) endpointsRealizer)) adjacentRealizer |
| def uRealizer : Realizer u := composition (pair first.snd.fst second) (getDFast (0,1)) |
| def vRealizer : Realizer v := composition (pair first.snd.snd second) (getDFast (0,1)) |
| lemma polynomial_p : PolynomialTime pRealizer := by |
| unfold pRealizer p Realizer.fst Realizer.snd |
| have := polynomial_endpoints |
| have := polynomial_incident |
| tree_cost |
| lemma polynomial_q : PolynomialTime qRealizer := by |
| unfold qRealizer q Realizer.fst Realizer.snd |
| have := polynomial_endpoints |
| have := polynomial_same |
| tree_cost |
| lemma polynomial_r : PolynomialTime rRealizer := by |
| unfold rRealizer r Realizer.fst Realizer.snd |
| have := polynomial_endpoints |
| have := polynomial_same |
| tree_cost |
| lemma polynomial_t : PolynomialTime tRealizer := by |
| unfold tRealizer t Realizer.fst Realizer.snd |
| have := polynomial_endpoints |
| have := polynomial_adjacent |
| tree_cost |
| lemma polynomial_u : PolynomialTime uRealizer := |
| polynomial_composition |
| (polynomial_pair (polynomial_fst (polynomial_snd polynomial_first)) polynomial_second) |
| (polynomial_getDFast ((0,1) : Fraction)) |
| lemma polynomial_v : PolynomialTime vRealizer := |
| polynomial_composition |
| (polynomial_pair (polynomial_snd (polynomial_snd polynomial_first)) polynomial_second) |
| (polynomial_getDFast ((0,1) : Fraction)) |
|
|
| def draw := pairDraw p q r t u v |
| def drawRealizer : Realizer draw := pairDrawRealizer pRealizer qRealizer rRealizer tRealizer |
| uRealizer vRealizer natEqual leNat natEqual leNat |
| lemma polynomial_draw : PolynomialTime drawRealizer := |
| polynomial_pairDraw polynomial_p polynomial_q polynomial_r polynomial_t polynomial_u polynomial_v |
| polynomial_natEqual polynomial_leNat polynomial_natEqual polynomial_leNat |
| end MatchingFPRAS.ExecutablePair |
|
|
| end OAI |
|
|
