| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace ExtremalEdges | |
| open Finset | |
| variable {V I : Type*} [Fintype V] [Fintype I] | |
| attribute [local instance] Classical.propDecidable | |
| local instance capMatchingDecidableEq (carrier : Sort*) : DecidableEq carrier := | |
| Classical.typeDecidableEq carrier | |
| def edgeEnds (e : I → V × V) (i : I) : Finset V := {(e i).1,(e i).2} | |
| def incident (e : I → V × V) (x : V) : Finset I := | |
| Finset.univ.filter (fun i => (e i).1 = x ∨ (e i).2 = x) | |
| lemma card_incident (G : SimpleGraph V) [DecidableRel G.Adj] (D : ℕ) | |
| (hd : ∀ x, G.degree x ≤ D) (e : I → V × V) | |
| (he : ∀ i, G.Adj (e i).1 (e i).2) (hi : Function.Injective e) (x : V) : | |
| (incident e x).card ≤ 2*D := by | |
| let L := Finset.univ.filter (fun i => (e i).1 = x) | |
| let R := Finset.univ.filter (fun i => (e i).2 = x) | |
| have hL : L.card ≤ G.degree x := by | |
| rw [← G.card_neighborFinset_eq_degree] | |
| apply Finset.card_le_card_of_injOn (fun i => (e i).2) | |
| · intro i hiL | |
| have hix : (e i).1 = x := (Finset.mem_filter.mp hiL).2 | |
| simpa [← hix] using he i | |
| · intro i hiL k hkL hik | |
| apply hi | |
| exact Prod.ext ((Finset.mem_filter.mp hiL).2.trans (Finset.mem_filter.mp hkL).2.symm) hik | |
| have hR : R.card ≤ G.degree x := by | |
| rw [← G.card_neighborFinset_eq_degree] | |
| apply Finset.card_le_card_of_injOn (fun i => (e i).1) | |
| · intro i hiR | |
| have hix : (e i).2 = x := (Finset.mem_filter.mp hiR).2 | |
| simpa [← hix] using (he i).symm | |
| · intro i hiR k hkR hik | |
| apply hi | |
| exact Prod.ext hik ((Finset.mem_filter.mp hiR).2.trans (Finset.mem_filter.mp hkR).2.symm) | |
| have heq : incident e x = L ∪ R := by ext i; simp [incident,L,R] | |
| rw [heq] | |
| have hh := Finset.card_union_le L R | |
| have hdx := hd x | |
| omega | |
| theorem large_matching (G : SimpleGraph V) [DecidableRel G.Adj] (D : ℕ) | |
| (hd : ∀ x, G.degree x ≤ D) (e : I → V × V) | |
| (he : ∀ i, G.Adj (e i).1 (e i).2) (hi : Function.Injective e) : | |
| ∃ M : Finset I, (M : Set I).Pairwise (fun i k => Disjoint (edgeEnds e i) (edgeEnds e k)) ∧ | |
| Fintype.card I ≤ M.card*(4*D) := by | |
| let T : Finset (Finset I) := Finset.univ.filter | |
| (fun M => (M : Set I).Pairwise (fun i k => Disjoint (edgeEnds e i) (edgeEnds e k))) | |
| have hT : T.Nonempty := ⟨∅, by simp [T]⟩ | |
| obtain ⟨M,hmax⟩ := T.exists_maximal hT | |
| have hM : (M : Set I).Pairwise (fun i k => Disjoint (edgeEnds e i) (edgeEnds e k)) := | |
| (Finset.mem_filter.mp hmax.1).2 | |
| have cover (i : I) : ∃ m ∈ M, ¬Disjoint (edgeEnds e i) (edgeEnds e m) := by | |
| by_contra hn | |
| push Not at hn | |
| have hiM : i ∉ M := by | |
| intro hh | |
| exact Finset.disjoint_left.mp (hn i hh) | |
| (show (e i).1 ∈ edgeEnds e i by simp [edgeEnds]) (by simp [edgeEnds]) | |
| have hpair : ((insert i M : Finset I) : Set I).Pairwise (fun a b => Disjoint (edgeEnds e a) (edgeEnds e b)) := by | |
| intro a ha b hb hab | |
| rcases Finset.mem_insert.mp ha with hai | ham | |
| · subst a | |
| rcases Finset.mem_insert.mp hb with hbi | hbm | |
| · exact False.elim (hab hbi.symm) | |
| · exact hn b hbm | |
| · rcases Finset.mem_insert.mp hb with hbi | hbm | |
| · subst b | |
| exact (hn a ham).symm | |
| · exact hM ham hbm hab | |
| have hmem : insert i M ∈ T := by simp only [T,Finset.mem_filter,Finset.mem_univ,true_and]; exact hpair | |
| exact hiM ((hmax.2 hmem (Finset.subset_insert i M)) (by simp)) | |
| let touch (m : I) := incident e (e m).1 ∪ incident e (e m).2 | |
| have hall : (Finset.univ : Finset I) ⊆ M.biUnion touch := by | |
| intro i _ | |
| obtain ⟨m,hm,hmt⟩ := cover i | |
| refine Finset.mem_biUnion.mpr ⟨m,hm,?_⟩ | |
| have hs : ∃ x, x ∈ edgeEnds e i ∧ x ∈ edgeEnds e m := by | |
| simpa only [Finset.not_disjoint_iff] using hmt | |
| rcases hs with ⟨x,hx,hy⟩ | |
| simp only [edgeEnds, Finset.mem_insert, Finset.mem_singleton] at hx hy | |
| rcases hx with rfl | rfl <;> rcases hy with hy | hy <;> | |
| simp [touch,incident,hy] | |
| have ht (m : I) : (touch m).card ≤ 4*D := by | |
| have h1 := card_incident G D hd e he hi (e m).1 | |
| have h2 := card_incident G D hd e he hi (e m).2 | |
| have hh := Finset.card_union_le (incident e (e m).1) (incident e (e m).2) | |
| dsimp [touch] | |
| omega | |
| refine ⟨M,hM,?_⟩ | |
| simpa only [Finset.card_univ] using (Finset.card_le_card hall).trans | |
| (Finset.card_biUnion_le_card_mul M touch (4*D) (fun m _ => ht m)) | |
| end ExtremalEdges | |
| end | |
| end OAI | |