File size: 2,786 Bytes
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 | 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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