| import OAI.Combinatorics.PortMatching.Basic |
|
|
| namespace OAI |
|
|
| /-! Perfect-matching-preserving port-path degree reduction for finite simple graphs. -/ |
|
|
| noncomputable section |
| namespace Matching205.Forward |
| universe u |
|
|
| def Pair (c i j : ℕ) : Prop := |
| (i < c ∧ i % 2 = 0 ∧ j = i + 1) ∨ |
| (j < c ∧ j % 2 = 0 ∧ i = j + 1) ∨ |
| (c < i ∧ i % 2 = 1 ∧ j = i + 1) ∨ |
| (c < j ∧ j % 2 = 1 ∧ i = j + 1) |
|
|
| lemma pair_symm {c i j : ℕ} (h : Pair c i j) : Pair c j i := by |
| rcases h with h | h | h | h |
| · exact Or.inr (Or.inl h) |
| · exact Or.inl h |
| · exact Or.inr (Or.inr (Or.inr h)) |
| · exact Or.inr (Or.inr (Or.inl h)) |
|
|
| lemma pair_consecutive {c i j : ℕ} (h : Pair c i j) : |
| i + 1 = j ∨ j + 1 = i := by |
| dsimp [Pair] at h |
| omega |
|
|
| lemma pair_not_center {k i j : ℕ} (h : Pair (2 * k) i j) : i ≠ 2 * k := by |
| dsimp [Pair] at h |
| omega |
|
|
| lemma pair_completion (r k : ℕ) (hk : k < r) |
| (i : Fin (2 * r - 1)) (hi : i.val ≠ 2 * k) : |
| ∃! j : Fin (2 * r - 1), Pair (2 * k) i.val j.val := by |
| have hib := i.isLt |
| by_cases hl : i.val < 2 * k |
| · by_cases hp : i.val % 2 = 0 |
| · refine ⟨⟨i.val + 1, by omega⟩, ?_, ?_⟩ |
| · dsimp [Pair]; omega |
| · intro j hj; apply Fin.ext; dsimp [Pair] at hj; dsimp; omega |
| · refine ⟨⟨i.val - 1, by omega⟩, ?_, ?_⟩ |
| · dsimp [Pair]; omega |
| · intro j hj; apply Fin.ext; dsimp [Pair] at hj; dsimp; omega |
| · by_cases hp : i.val % 2 = 0 |
| · refine ⟨⟨i.val - 1, by omega⟩, ?_, ?_⟩ |
| · dsimp [Pair]; omega |
| · intro j hj; apply Fin.ext; dsimp [Pair] at hj; dsimp; omega |
| · refine ⟨⟨i.val + 1, by omega⟩, ?_, ?_⟩ |
| · dsimp [Pair]; omega |
| · intro j hj; apply Fin.ext; dsimp [Pair] at hj; dsimp; omega |
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| variable {V : Type u} [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj] |
| variable (M : G.Subgraph) (hM : M.IsPerfectMatching) |
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|
| omit [Fintype V] [DecidableRel G.Adj] in |
| include hM in |
| private lemma unique_neighbor (v : V) : ∃! w, M.Adj v w := |
| SimpleGraph.Subgraph.isPerfectMatching_iff.mp hM v |
|
|
| def neighbor (v : V) : G.neighborSet v := |
| ⟨(unique_neighbor G M hM v).choose, |
| M.adj_sub (unique_neighbor G M hM v).choose_spec.1⟩ |
|
|
| omit [Fintype V] [DecidableRel G.Adj] in |
| lemma neighbor_adj (v : V) : M.Adj v (neighbor G M hM v).val := |
| (unique_neighbor G M hM v).choose_spec.1 |
|
|
| def center (v : V) : ℕ := |
| 2 * ((Fintype.equivFin (G.neighborSet v)) (neighbor G M hM v)).val |
|
|
| def selected (v : V) : Vertex G := port G v (neighbor G M hM v) |
|
|
| lemma port_eq_selected {v w : V} (h : M.Adj v w) : |
| port G v ⟨w, M.adj_sub h⟩ = selected G M hM v := by |
| have hw : w = (neighbor G M hM v).val := |
| (unique_neighbor G M hM v).choose_spec.2 w h |
| have hn : (⟨w, M.adj_sub h⟩ : G.neighborSet v) = neighbor G M hM v := |
| Subtype.ext hw |
| exact congrArg (port G v) hn |
|
|
| def Local (x y : Vertex G) : Prop := |
| x.1 = y.1 ∧ Pair (center G M hM x.1) x.2.val y.2.val |
|
|
| def Across (x y : Vertex G) : Prop := |
| External G x y ∧ M.Adj x.1 y.1 |
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| lemma local_symm {x y : Vertex G} (h : Local G M hM x y) : Local G M hM y x := by |
| refine ⟨h.1.symm, ?_⟩ |
| have hc : center G M hM x.1 = center G M hM y.1 := |
| congrArg (center G M hM) h.1 |
| have hp : Pair (center G M hM x.1) y.2.val x.2.val = |
| Pair (center G M hM y.1) y.2.val x.2.val := |
| congrArg (fun c : ℕ => Pair c y.2.val x.2.val) hc |
| exact hp.mp (pair_symm h.2) |
|
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| lemma across_symm {x y : Vertex G} (h : Across G M x y) : Across G M y x := |
| ⟨external_symm G h.1, h.2.symm⟩ |
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| lemma across_selected {x y : Vertex G} (h : Across G M x y) : |
| x = selected G M hM x.1 := by |
| obtain ⟨hg, hx, _⟩ := h.1 |
| exact hx.trans (port_eq_selected G M hM h.2) |
|
|
| include hM in |
| lemma across_unique {x y z : Vertex G} |
| (hy : Across G M x y) (hz : Across G M x z) : y = z := by |
| have hv : y.1 = z.1 := hM.1.eq_of_adj_left hy.2 hz.2 |
| calc |
| y = selected G M hM y.1 := across_selected G M hM (across_symm G M hy) |
| _ = selected G M hM z.1 := congrArg (selected G M hM) hv |
| _ = z := (across_selected G M hM (across_symm G M hz)).symm |
|
|
| lemma selected_across (v : V) : ∃ y, Across G M (selected G M hM v) y := by |
| let w := neighbor G M hM v |
| let h : G.Adj v w.val := w.property |
| refine ⟨port G w.val ⟨v, h.symm⟩, ?_⟩ |
| exact ⟨⟨h, rfl, rfl⟩, neighbor_adj G M hM v⟩ |
|
|
| lemma local_not_selected {x y : Vertex G} (h : Local G M hM x y) : |
| x ≠ selected G M hM x.1 := by |
| intro hx |
| have hp := pair_not_center h.2 |
| apply hp |
| have hv := congrArg (fun z : Vertex G => z.2.val) hx |
| exact hv |
|
|
| lemma local_completion (x : Vertex G) (hx : x ≠ selected G M hM x.1) : |
| ∃! y, Local G M hM x y := by |
| rcases x with ⟨v, i⟩ |
| let k := (Fintype.equivFin (G.neighborSet v)) (neighbor G M hM v) |
| have hi : i.val ≠ 2 * k.val := by |
| intro hi |
| apply hx |
| have hfin : i = (selected G M hM v).2 := Fin.ext hi |
| cases hfin |
| rfl |
| obtain ⟨j, hj, hu⟩ := pair_completion (arity G v) k.val k.isLt i hi |
| refine ⟨⟨v, j⟩, ⟨rfl, hj⟩, ?_⟩ |
| rintro ⟨w, l⟩ ⟨hw, hl⟩ |
| dsimp at hw |
| subst w |
| have hlj : l = j := hu l hl |
| cases hlj |
| rfl |
|
|
| def extend : (graph G).Subgraph where |
| verts := Set.univ |
| Adj x y := Local G M hM x y ∨ Across G M x y |
| adj_sub := by |
| intro x y h |
| rcases h with h | h |
| · exact Or.inl ⟨h.1, pair_consecutive h.2⟩ |
| · exact Or.inr h.1 |
| edge_vert := fun _ => Set.mem_univ _ |
| symm := ⟨by |
| intro x y h |
| exact h.elim (fun h => Or.inl (local_symm G M hM h)) |
| (fun h => Or.inr (across_symm G M h))⟩ |
|
|
| theorem extend_perfect : (extend G M hM).IsPerfectMatching := by |
| classical |
| apply SimpleGraph.Subgraph.isPerfectMatching_iff.mpr |
| intro x |
| by_cases hx : x = selected G M hM x.1 |
| · obtain ⟨y, hy⟩ := selected_across G M hM x.1 |
| have hy' : Across G M x y := hx.symm ▸ hy |
| refine ⟨y, Or.inr hy', ?_⟩ |
| intro z hz |
| rcases hz with hz | hz |
| · exact False.elim (local_not_selected G M hM hz hx) |
| · exact across_unique G M hM hz hy' |
| · obtain ⟨y, hy, hu⟩ := local_completion G M hM x hx |
| refine ⟨y, Or.inl hy, ?_⟩ |
| intro z hz |
| rcases hz with hz | hz |
| · exact hu z hz |
| · exact False.elim (hx (across_selected G M hM hz)) |
|
|
| theorem decode_extend_adj (v w : V) : |
| (decode G (extend G M hM)).Adj v w ↔ M.Adj v w := by |
| constructor |
| · rintro ⟨h, hm⟩ |
| rcases hm with hi | he |
| · have hvw : v = w := hi.1 |
| subst w |
| exact False.elim (G.irrefl h) |
| · exact he.2 |
| · intro h |
| exact ⟨M.adj_sub h, Or.inr ⟨⟨M.adj_sub h, rfl, rfl⟩, h⟩⟩ |
|
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| end Matching205.Forward |
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| end |
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| end OAI |
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