| import OAI.Combinatorics.PortMatching.Basic |
| import Mathlib.Combinatorics.SimpleGraph.DegreeSum |
|
|
| namespace OAI |
|
|
| /-! Perfect-matching-preserving port-path degree reduction for finite simple graphs. -/ |
|
|
| noncomputable section |
| namespace Matching205 |
|
|
| universe u |
| variable {V : Type u} [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj] |
|
|
| noncomputable instance graphAdjDecidable : DecidableRel (graph G).Adj := |
| fun _ _ => Classical.propDecidable _ |
|
|
| private lemma size_vertex_ext {x y : Vertex G} |
| (hbase : x.1 = y.1) (hpos : x.2.val = y.2.val) : x = y := by |
| rcases x with ⟨v, i⟩ |
| rcases y with ⟨w, j⟩ |
| dsimp at hbase hpos |
| subst w |
| congr 1 |
| exact Fin.ext hpos |
|
|
| private lemma size_port_injective (v : V) : Function.Injective (port G v) := by |
| intro a b hab |
| have h := congrArg (fun x : Vertex G => x.2.val) hab |
| change 2 * ((Fintype.equivFin (G.neighborSet v)) a).val = |
| 2 * ((Fintype.equivFin (G.neighborSet v)) b).val at h |
| apply (Fintype.equivFin (G.neighborSet v)).injective |
| apply Fin.ext |
| omega |
|
|
| private lemma size_external_base_ne {x y : Vertex G} (h : External G x y) : |
| x.1 ≠ y.1 := by |
| obtain ⟨h, _, _⟩ := h |
| exact h.ne |
|
|
| private lemma size_external_unique {x y z : Vertex G} |
| (hy : External G x y) (hz : External G x z) : y = z := by |
| obtain ⟨hxy, hx, hy⟩ := hy |
| obtain ⟨hxz, hx', hz⟩ := hz |
| have hn := size_port_injective G x.1 (hx.symm.trans hx') |
| have hbase : y.1 = z.1 := congrArg Subtype.val hn |
| have transport (v w : V) (h : v = w) (hv : G.Adj v x.1) (hw : G.Adj w x.1) : |
| port G v ⟨x.1, hv⟩ = port G w ⟨x.1, hw⟩ := by |
| subst w |
| rfl |
| have hp := transport y.1 z.1 hbase hxy.symm hxz.symm |
| exact hy.trans (hp.trans hz.symm) |
|
|
| private def size_neighborCode (x : Vertex G) (y : (graph G).neighborSet x) : Fin 3 := by |
| classical |
| exact if y.val.1 = x.1 then |
| if y.val.2.val + 1 = x.2.val then 0 else 1 |
| else 2 |
|
|
| private lemma size_neighborCode_injective (x : Vertex G) : |
| Function.Injective (size_neighborCode G x) := by |
| classical |
| intro y z hc |
| have hy : Internal G x y.val ∨ External G x y.val := y.property |
| have hz : Internal G x z.val ∨ External G x z.val := z.property |
| apply Subtype.ext |
| rcases hy with hy | hy <;> rcases hz with hz | hz |
| · apply size_vertex_ext G (hy.1.symm.trans hz.1) |
| have hv := congrArg Fin.val hc |
| by_cases hyp : y.val.2.val + 1 = x.2.val <;> |
| by_cases hzp : z.val.2.val + 1 = x.2.val |
| all_goals simp [size_neighborCode, hy.1.symm, hz.1.symm, hyp, hzp] at hv |
| all_goals rcases hy.2 with hy' | hy' <;> rcases hz.2 with hz' | hz' <;> omega |
| · have hn : z.val.1 ≠ x.1 := (size_external_base_ne G hz).symm |
| have hv := congrArg Fin.val hc |
| by_cases hp : y.val.2.val + 1 = x.2.val <;> |
| simp [size_neighborCode, hy.1.symm, hn, hp] at hv |
| · have hn : y.val.1 ≠ x.1 := (size_external_base_ne G hy).symm |
| have hv := congrArg Fin.val hc |
| by_cases hp : z.val.2.val + 1 = x.2.val <;> |
| simp [size_neighborCode, hz.1.symm, hn, hp] at hv |
| · exact size_external_unique G hy hz |
|
|
| theorem graph_degree_le_three (x : Vertex G) : (graph G).degree x ≤ 3 := by |
| classical |
| rw [← SimpleGraph.card_neighborSet_eq_degree] |
| simpa using Fintype.card_le_of_injective (size_neighborCode G x) |
| (size_neighborCode_injective G x) |
|
|
| theorem card_vertex_add_card (hpos : ∀ v : V, 0 < G.degree v) : |
| Fintype.card (Vertex G) + Fintype.card V = 4 * G.edgeFinset.card := by |
| classical |
| have hfiber (v : V) : Fintype.card (Fin (2 * arity G v - 1)) + 1 = |
| 2 * G.degree v := by |
| rw [card_fiber G v] |
| have hv := hpos v |
| omega |
| calc |
| Fintype.card (Vertex G) + Fintype.card V = |
| ∑ v : V, (Fintype.card (Fin (2 * arity G v - 1)) + 1) := by |
| simp [Vertex, Fintype.card_sigma, Finset.sum_add_distrib] |
| _ = ∑ v : V, 2 * G.degree v := Finset.sum_congr rfl (fun v _ => hfiber v) |
| _ = 2 * ∑ v : V, G.degree v := (Finset.mul_sum _ _ _).symm |
| _ = 4 * G.edgeFinset.card := by |
| rw [G.sum_degrees_eq_twice_card_edges] |
| omega |
|
|
| theorem card_vertex (hpos : ∀ v : V, 0 < G.degree v) : |
| Fintype.card (Vertex G) = 4 * G.edgeFinset.card - Fintype.card V := by |
| have h := card_vertex_add_card G hpos |
| omega |
|
|
| end Matching205 |
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| end |
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| end OAI |
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|
