| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Graphs.LabelOrder |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.RationalSource |
| open Finset Transport Enlargement Subdivision RealGeometry Switching |
|
|
| variable {n : ℕ} |
| def pairIndex (he : Even n) (i : Fin n) : Fin n := |
| if hi : i.val%2=0 then ⟨i.val+1,by obtain ⟨k,hk⟩:=he; have h:=i.isLt; omega⟩ |
| else ⟨i.val-1,by have h:=i.isLt; omega⟩ |
| lemma pairIndex_involutive (he : Even n) : Function.Involutive (pairIndex he) := by |
| intro i |
| apply Fin.ext |
| by_cases hi : i.val % 2 = 0 |
| · have hp : (i.val+1) % 2 ≠ 0 := by omega |
| simp only [pairIndex, dite_eq_left hi, dite_eq_right hp] |
| omega |
| · have hp : (i.val-1) % 2 = 0 := by omega |
| simp only [pairIndex, dite_eq_right hi, dite_eq_left hp] |
| omega |
| lemma pairIndex_ne (he : Even n) (i : Fin n) : pairIndex he i≠i := by |
| intro h |
| have hh:=congrArg Fin.val h |
| unfold pairIndex at hh |
| split_ifs at hh <;> simp only at hh <;> omega |
|
|
| def orderedMate (he : Even n) : Mate (Fin n) := ⟨pairIndex he,pairIndex_involutive he⟩ |
| lemma orderedMate_holes (he : Even n) : HasHoles (orderedMate he) ∅ := by |
| intro i |
| simp only [Finset.notMem_empty,iff_false] |
| exact pairIndex_ne he i |
|
|
| def selectMate (m : Mate (Fin n)) (e : LogicalEdge (Fin n)) : Bool := |
| decide (m.val (source e)=target e) |
| lemma selectMate_eq (m : Mate (Fin n)) : selectMate m=Representation.select m := by |
| funext e |
| exact rfl |
|
|
| def startSet (K : ℕ) (he : Even n) : Finset (SourceRealEdge (Fin n) K) := |
| univ.filter fun q=>alternating (selectMate (orderedMate he) q.1) q.2.val=true |
| variable [Nontrivial (Fin n)] |
| lemma startSet_matching (K : ℕ) (he : Even n) : |
| MatchingOn (fun (q : SourceRealEdge (Fin n) K) x=>x∈edge source target (4*(2*K+2)) q.1 q.2) |
| (startSet K he) univ := by |
| have h:= (family_iff_matching source target (4*(2*K+2)) source_ne_target ∅ (fun _=>∅) |
| (fun e i=>alternating (Representation.select (orderedMate he) e) i.val)).mp |
| (intact_family (2*K+2) (orderedMate he) (orderedMate_holes he)) |
| rw [survivors_empty] at h |
| convert h using 1 |
| unfold startSet selectedSet |
| rw [selectMate_eq] |
|
|
| attribute [local instance] NumberedHierarchy.labelOrder sym2Order bagEdgeOrder bagEdgeFintype |
| attribute [local instance] UnitRefresh.Computable.allowedDec prodOrder sumOrder |
|
|
| def startMatching (w : Fin n → Fin n → ℚ) (K : ℕ) (he : Even n) : BagPM (bags w K) := |
| ⟨(startSet K he).map ⟨coloredReal w K,coloredReal_injective w K⟩,by |
| exact (matching_map_iff |
| (fun (q : SourceRealEdge (Fin n) K) x=>x∈edge source target (4*(2*K+2)) q.1 q.2) |
| BagEdge.Incident ⟨coloredReal w K,coloredReal_injective w K⟩ |
| (fun _ _=>Iff.rfl) (startSet K he) univ).mpr (startSet_matching K he)⟩ |
| end MatchingFPRAS.RationalSource |
|
|
| end OAI |
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|
