| import OAI.Combinatorics.PortMatching.Basic |
|
|
| namespace OAI |
|
|
| /-! Perfect-matching-preserving port-path degree reduction for finite simple graphs. -/ |
|
|
| noncomputable section |
| namespace Matching205 |
| namespace Reverse |
| universe u |
| variable {V : Type u} [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj] |
|
|
| lemma unique_missing {r : ℕ} (hr : 0 < r) (f : Fin (r-1) → Fin r) |
| (hf : Function.Injective f) : ∃! p, ∀ q, f q ≠ p := by |
| classical |
| let s := Finset.univ.image f |
| have hs : s.card = r-1 := by |
| dsimp [s] |
| rw [Finset.card_image_of_injective _ hf, Finset.card_univ, Fintype.card_fin] |
| have hc : (Finset.univ \ s).card = 1 := by |
| rw [Finset.card_sdiff_of_subset (Finset.subset_univ s), Finset.card_univ, |
| Fintype.card_fin, hs] |
| omega |
| have h := Finset.card_eq_one_iff_existsUnique.mp hc |
| simpa [s, Finset.mem_sdiff, Finset.mem_image] using h |
|
|
| variable {G} |
| def mate {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) (x : Vertex G) : Vertex G := |
| (SimpleGraph.Subgraph.isPerfectMatching_iff.mp hM x).choose |
| lemma mate_adj {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) (x : Vertex G) : |
| M.Adj x (mate hM x) := |
| (SimpleGraph.Subgraph.isPerfectMatching_iff.mp hM x).choose_spec.1 |
| lemma mate_eq {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) |
| {x y : Vertex G} (h : M.Adj x y) : mate hM x = y := |
| hM.1.eq_of_adj_left (mate_adj hM x) h |
| lemma mate_involutive {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) : |
| Function.Involutive (mate hM) := fun x => mate_eq hM (mate_adj hM x).symm |
|
|
| variable (G) |
| def evenPosition (hpos : ∀ v, 0 < arity G v) (v : V) (j : Fin (arity G v)) : Vertex G := |
| ⟨v, ⟨2*j.val, by have := j.isLt; have := hpos v; omega⟩⟩ |
| def nonport (v : V) (q : Fin (arity G v-1)) : Vertex G := |
| ⟨v, ⟨2*q.val+1, by have := q.isLt; omega⟩⟩ |
| lemma evenPosition_injective (hpos : ∀ v, 0 < arity G v) (v : V) : |
| Function.Injective (evenPosition G hpos v) := by |
| intro i j h |
| have h' := congrArg (fun x : Vertex G => x.2.val) h |
| apply Fin.ext |
| dsimp [evenPosition] at h' |
| omega |
| lemma nonport_injective (v : V) : Function.Injective (nonport G v) := by |
| intro i j h |
| have h' := congrArg (fun x : Vertex G => x.2.val) h |
| apply Fin.ext |
| dsimp [nonport] at h' |
| omega |
| lemma port_eq_evenPosition (hpos : ∀ v, 0 < arity G v) (v : V) |
| (w : G.neighborSet v) : |
| port G v w = evenPosition G hpos v (Fintype.equivFin _ w) := rfl |
|
|
| lemma nonport_not_external (v : V) (q : Fin (arity G v-1)) (y : Vertex G) : |
| ¬ External G (nonport G v q) y := by |
| rintro ⟨h, hx, _⟩ |
| have hx' := congrArg (fun x : Vertex G => x.2.val) hx |
| dsimp [nonport, port] at hx' |
| omega |
|
|
| lemma internal_nonport_even (hpos : ∀ v, 0 < arity G v) (v : V) |
| (q : Fin (arity G v-1)) {y : Vertex G} |
| (h : Internal G (nonport G v q) y) : |
| ∃ j, y = evenPosition G hpos v j := by |
| rcases y with ⟨w,i⟩ |
| have hw : v = w := h.1 |
| subst w |
| have hi := i.isLt |
| have hn := h.2 |
| dsimp [nonport] at hn |
| have he : i.val % 2 = 0 := by omega |
| let j : Fin (arity G v) := ⟨i.val / 2, by omega⟩ |
| refine ⟨j, ?_⟩ |
| congr 1 |
| apply Fin.ext |
| dsimp [evenPosition, j] |
| omega |
|
|
| lemma internal_even_nonport (hpos : ∀ v, 0 < arity G v) (v : V) |
| (j : Fin (arity G v)) {y : Vertex G} |
| (h : Internal G (evenPosition G hpos v j) y) : |
| ∃ q, y = nonport G v q := by |
| rcases y with ⟨w,i⟩ |
| have hw : v = w := h.1 |
| subst w |
| have hi := i.isLt |
| have hn := h.2 |
| dsimp [evenPosition] at hn |
| have he : i.val % 2 = 1 := by omega |
| let q : Fin (arity G v-1) := ⟨i.val / 2, by omega⟩ |
| refine ⟨q, ?_⟩ |
| congr 1 |
| apply Fin.ext |
| dsimp [nonport, q] |
| omega |
|
|
| lemma mate_nonport_even (hpos : ∀ v, 0 < arity G v) |
| {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) |
| (v : V) (q : Fin (arity G v-1)) : |
| ∃ j, mate hM (nonport G v q) = evenPosition G hpos v j := by |
| have h := M.adj_sub (mate_adj hM (nonport G v q)) |
| change Internal G _ _ ∨ External G _ _ at h |
| rcases h with h | h |
| · exact internal_nonport_even G hpos v q h |
| · exact (nonport_not_external G v q _ h).elim |
|
|
| def pairedPort (hpos : ∀ v, 0 < arity G v) |
| {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) |
| (v : V) (q : Fin (arity G v-1)) : Fin (arity G v) := |
| (mate_nonport_even G hpos hM v q).choose |
| lemma pairedPort_spec (hpos : ∀ v, 0 < arity G v) |
| {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) |
| (v : V) (q : Fin (arity G v-1)) : |
| mate hM (nonport G v q) = evenPosition G hpos v (pairedPort G hpos hM v q) := |
| (mate_nonport_even G hpos hM v q).choose_spec |
| lemma pairedPort_injective (hpos : ∀ v, 0 < arity G v) |
| {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) (v : V) : |
| Function.Injective (pairedPort G hpos hM v) := by |
| intro i j h |
| apply nonport_injective G v |
| apply (mate_involutive hM).injective |
| rw [pairedPort_spec G hpos hM, pairedPort_spec G hpos hM, h] |
|
|
| lemma external_iff_missing (hpos : ∀ v, 0 < arity G v) |
| {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) |
| (v : V) (j : Fin (arity G v)) : |
| External G (evenPosition G hpos v j) (mate hM (evenPosition G hpos v j)) ↔ |
| ∀ q, pairedPort G hpos hM v q ≠ j := by |
| constructor |
| · intro he q hq |
| have hp := pairedPort_spec G hpos hM v q |
| rw [hq] at hp |
| have hm := congrArg (mate hM) hp |
| rw [mate_involutive hM] at hm |
| obtain ⟨ha, _, _⟩ := he |
| rw [← hm] at ha |
| exact G.irrefl ha |
| · intro hj |
| have h := M.adj_sub (mate_adj hM (evenPosition G hpos v j)) |
| change Internal G _ _ ∨ External G _ _ at h |
| rcases h with hi | he |
| · obtain ⟨q, hq⟩ := internal_even_nonport G hpos v j hi |
| have hm := congrArg (mate hM) hq |
| rw [mate_involutive hM, pairedPort_spec G hpos hM] at hm |
| exact (hj q ((evenPosition_injective G hpos v hm).symm)).elim |
| · exact he |
|
|
| theorem unique_external_port (hpos : ∀ v, 0 < arity G v) |
| {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) (v : V) : |
| ∃! j : Fin (arity G v), |
| External G (evenPosition G hpos v j) (mate hM (evenPosition G hpos v j)) := by |
| have h := unique_missing (hpos v) (pairedPort G hpos hM v) |
| (pairedPort_injective G hpos hM v) |
| simpa only [external_iff_missing] using h |
|
|
| end Reverse |
|
|
| universe u |
| variable {V : Type u} [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj] |
|
|
| theorem decode_isPerfectMatching (hpos : ∀ v, 0 < arity G v) |
| {M : (graph G).Subgraph} (hM : M.IsPerfectMatching) : |
| (decode G M).IsPerfectMatching := by |
| classical |
| apply SimpleGraph.Subgraph.isPerfectMatching_iff.mpr |
| intro v |
| obtain ⟨j, hj, hu⟩ := Reverse.unique_external_port G hpos hM v |
| obtain ⟨ha, hx, hy⟩ := hj |
| let y := Reverse.mate hM (Reverse.evenPosition G hpos v j) |
| have ha' : G.Adj v y.1 := ha |
| have hdec : (decode G M).Adj v y.1 := by |
| refine ⟨ha', ?_⟩ |
| have hm := Reverse.mate_adj hM (Reverse.evenPosition G hpos v j) |
| exact (congrArg₂ (fun a b : Vertex G => M.Adj a b) hx hy).mp hm |
| refine ⟨y.1, hdec, ?_⟩ |
| intro w hw |
| obtain ⟨haw, hmw⟩ := hw |
| let k : Fin (arity G v) := Fintype.equivFin _ ⟨w, haw⟩ |
| have hport : port G v ⟨w, haw⟩ = Reverse.evenPosition G hpos v k := rfl |
| have hmate : Reverse.mate hM (Reverse.evenPosition G hpos v k) = |
| port G w ⟨v, haw.symm⟩ := by |
| rw [← hport] |
| exact Reverse.mate_eq hM hmw |
| have hk : External G (Reverse.evenPosition G hpos v k) |
| (Reverse.mate hM (Reverse.evenPosition G hpos v k)) := by |
| rw [hmate, ← hport] |
| exact ⟨haw, rfl, rfl⟩ |
| have hkj : k = j := hu k hk |
| have heq : port G v ⟨w, haw⟩ = port G v ⟨y.1, ha'⟩ := by |
| rw [hport, hkj] |
| exact hx |
| have hi := congrArg (fun x : Vertex G => x.2.val) heq |
| dsimp [port] at hi |
| have hf : Fintype.equivFin (G.neighborSet v) ⟨w, haw⟩ = |
| Fintype.equivFin (G.neighborSet v) ⟨y.1, ha'⟩ := by |
| apply Fin.ext |
| omega |
| exact congrArg Subtype.val ((Fintype.equivFin _).injective hf) |
|
|
| end Matching205 |
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| end |
|
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| end OAI |
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