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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 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import OAI.Combinatorics.MatchingPSD.FiniteFactorization
namespace OAI
namespace PerfectMatchingPSD
open MatchingSmoothing
attribute [local instance] Classical.propDecidable
noncomputable def pairingEdge {n : ℕ} (M : Pairing (Fin n)) (v : Fin n) : Edge n :=
if h : v < M.1 v then ⟨(v,M.1 v),h⟩ else
⟨(M.1 v,v),lt_of_le_of_ne (le_of_not_gt h) (M.partner_ne v)⟩
lemma pairingEdge_partner {n : ℕ} (M : Pairing (Fin n)) (v : Fin n) :
pairingEdge M (M.1 v) = pairingEdge M v := by
by_cases h : v < M.1 v
· simp [pairingEdge, h, M.partner_partner, not_lt.mpr h.le]
· have h' : M.1 v < v := lt_of_le_of_ne (le_of_not_gt h) (M.partner_ne v)
simp [pairingEdge, h, h', M.partner_partner]
lemma pairingEdge_incident {n : ℕ} (M : Pairing (Fin n)) (v : Fin n) :
(pairingEdge M v).1.1 = v ∨ (pairingEdge M v).1.2 = v := by
unfold pairingEdge
split_ifs <;> simp
lemma pairingEdge_mem {n : ℕ} (M : Pairing (Fin n)) (v : Fin n) :
M.1 (pairingEdge M v).1.1 = (pairingEdge M v).1.2 := by
unfold pairingEdge
split_ifs <;> simp
lemma eq_pairingEdge {n : ℕ} (M : Pairing (Fin n)) (e : Edge n) (hM : M.1 e.1.1 = e.1.2)
(v : Fin n) (hv : e.1.1 = v ∨ e.1.2 = v) : e = pairingEdge M v := by
have he : e = pairingEdge M e.1.1 := by
simp only [pairingEdge, hM, e.2, dite_true]
rcases hv with rfl | rfl
· exact he
· rw [← hM, pairingEdge_partner]
exact he
noncomputable def fromPairing {n : ℕ} (M : Pairing (Fin n)) : PerfectMatching n :=
⟨Finset.univ.filter (fun e : Edge n => M.1 e.1.1 = e.1.2), by
intro v
apply Finset.card_eq_one.mpr
refine ⟨pairingEdge M v, ?_⟩
ext e
simp only [Finset.mem_filter, Finset.mem_univ, true_and, Finset.mem_singleton]
constructor
· rintro ⟨hM,hv⟩
exact eq_pairingEdge M e hM v hv
· rintro rfl
exact ⟨pairingEdge_mem M v, pairingEdge_incident M v⟩⟩
@[simp] lemma mem_fromPairing {n : ℕ} (M : Pairing (Fin n)) (e : Edge n) :
e ∈ (fromPairing M).1 ↔ M.1 e.1.1 = e.1.2 := by
simp [fromPairing]
lemma pairingEdge_cross {n : ℕ} (M : Pairing (Fin n)) (U : Finset (Fin n))
(v : Fin n) (hv : v ∈ U) (hp : M.1 v ∉ U) : Crosses U (pairingEdge M v) := by
unfold pairingEdge
split_ifs <;> simp [Crosses, hv, hp]
lemma cross_edge_inside {n : ℕ} (M : Pairing (Fin n)) (U : Finset (Fin n))
(e : Edge n) (he : e ∈ (fromPairing M).1) (hc : Crosses U e) :
∃ v : Fin n, (v ∈ U ∧ M.1 v ∉ U) ∧ pairingEdge M v = e := by
have hm := (mem_fromPairing M e).mp he
rcases hc with hc | hc
· refine ⟨e.1.1,⟨hc.1,?_⟩,(eq_pairingEdge M e hm _ (Or.inl rfl)).symm⟩
simpa only [hm] using hc.2
· refine ⟨e.1.2,⟨hc.2,?_⟩,(eq_pairingEdge M e hm _ (Or.inr rfl)).symm⟩
simpa only [← hm, M.partner_partner] using hc.1
lemma pairingEdge_inside_injective {n : ℕ} (M : Pairing (Fin n)) (U : Finset (Fin n)) :
Function.Injective (fun v : {v : Fin n // v ∈ U ∧ M.1 v ∉ U} => pairingEdge M v.1) := by
intro v w h
apply Subtype.ext
have hv := pairingEdge_incident M v.1
have hw := pairingEdge_incident M w.1
change pairingEdge M v.1 = pairingEdge M w.1 at h
rw [← h] at hw
rcases hv with hv | hv <;> rcases hw with hw | hw
· exact hv.symm.trans hw
· have hm := pairingEdge_mem M v.1
rw [hv,hw] at hm
exact False.elim (v.2.2 (hm ▸ w.2.1))
· have hm := pairingEdge_mem M v.1
rw [hw,hv] at hm
exact False.elim (w.2.2 (hm ▸ v.2.1))
· exact hv.symm.trans hw
lemma crossingCount_fromPairing {n : ℕ} (M : Pairing (Fin n)) (U : Finset (Fin n)) :
crossingCount U (fromPairing M) = Fintype.card {v : Fin n // v ∈ U ∧ M.1 v ∉ U} := by
classical
rw [crossingCount, ← Fintype.card_coe]
symm
apply Fintype.card_of_bijective (f := fun v : {v : Fin n // v ∈ U ∧ M.1 v ∉ U} =>
(⟨pairingEdge M v.1,by simp [pairingEdge_mem, pairingEdge_cross M U v.1 v.2.1 v.2.2]⟩ :
{e // e ∈ (fromPairing M).1.filter (Crosses U)}))
constructor
· intro v w h
exact pairingEdge_inside_injective M U (congrArg Subtype.val h)
· intro e
obtain ⟨v,hv,he⟩ := cross_edge_inside M U e.1 (Finset.mem_filter.mp e.2).1 (Finset.mem_filter.mp e.2).2
exact ⟨⟨v,hv⟩,Subtype.ext he⟩
lemma crossingCount_tableCoordinate {n : ℕ} (M : Pairing (Fin n)) (U : Finset (Fin n)) :
(crossingCount U (fromPairing M) : ℝ) =
tableCoordinate (fun v => decide (v ∈ U)) (true,false) M := by
rw [crossingCount_fromPairing]
simp [tableCoordinate, Fintype.card_subtype, Finset.sum_boole]
end PerfectMatchingPSD
end OAI
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