| import Mathlib |
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| namespace OAI |
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| section |
| section |
| section |
| noncomputable section |
| namespace CannonForest |
| open Set |
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| theorem edges_confined_to_core {V : Type*} [Finite V] (G : SimpleGraph V) |
| (hG : G.IsAcyclic) (S : Set V) (hS : (G.induce S).Preconnected) |
| (hother : ∀ x∉S,∀ y,G.Adj x y → ∃ z,G.Adj x z ∧ z≠y) : |
| ∀ x y,G.Adj x y → x∈S ∧ y∈S := by |
| classical |
| let R : SimpleGraph V := { |
| Adj := fun x y=>G.Adj x y ∧ ¬(x∈S ∧ y∈S) |
| symm := ⟨fun x y h=>⟨h.1.symm,fun hh=>h.2 hh.symm⟩⟩ |
| loopless := ⟨fun x h=>h.1.ne rfl⟩ } |
| have hRG : R≤G := fun _ _ h=>h.1 |
| have acR := hG.anti hRG |
| have unique_core {x y:V} (hx:x∈S) (hy:y∈S) (hxy:R.Reachable x y) : x=y := by |
| by_contra hn |
| obtain ⟨p,hp⟩ := hxy.exists_isPath |
| obtain ⟨q,hq⟩ := (hS ⟨x,hx⟩ ⟨y,hy⟩).exists_isPath |
| let f : G.induce S →g G := ⟨Subtype.val,fun h=>h⟩ |
| have he : p.mapLe hRG=q.map f := Subtype.mk.inj |
| (hG.subsingleton_path x y |>.elim ⟨_,hp.mapLe hRG⟩ ⟨_,hq.map Subtype.val_injective⟩) |
| have hs : ∀ z∈p.support,z∈S := by |
| intro z hz |
| have hz' : z∈(p.mapLe hRG).support := by simpa using hz |
| rw [he,SimpleGraph.Walk.support_map] at hz' |
| obtain ⟨w,-,rfl⟩ := List.mem_map.mp hz' |
| exact w.property |
| have hpne : ¬p.Nil := SimpleGraph.Walk.not_nil_of_ne hn |
| exact (p.adj_snd hpne).2 ⟨hx,hs _ (List.mem_of_mem_tail (p.snd_mem_tail_support hpne))⟩ |
| have hempty : R=⊥ := by |
| ext x y |
| constructor |
| · intro hxy |
| let c := R.connectedComponentMk x |
| let a : c.supp := ⟨x,rfl⟩ |
| let b : c.supp := ⟨y,SimpleGraph.ConnectedComponent.sound hxy.reachable.symm⟩ |
| have hab : a≠b := fun h=>hxy.ne (congrArg Subtype.val h) |
| let _ : Nontrivial c.supp := ⟨⟨a,b,hab⟩⟩ |
| let _ : Fintype c.supp := Fintype.ofFinite c.supp |
| obtain ⟨u,v,huv,hu,hv⟩ := (acR.isTree_connectedComponent c).exists_ne_and_degree_eq_one |
| have leaves (z : c.supp) (hz : c.toSimpleGraph.degree z=1) : z.val∈S := by |
| by_contra hn |
| obtain ⟨w,hw,hunique⟩ := SimpleGraph.degree_eq_one_iff_existsUnique_adj.mp hz |
| have hwG : G.Adj z.val w.val := hRG hw |
| obtain ⟨t,ht,htw⟩ := hother z.val hn w.val hwG |
| have htR : R.Adj z.val t := ⟨ht,fun h=>hn h.1⟩ |
| let t' : c.supp := ⟨t,c.mem_supp_of_adj_mem_supp z.property htR⟩ |
| have hadj : c.toSimpleGraph.Adj z t' := htR |
| exact htw (congrArg Subtype.val (hunique _ hadj)) |
| have huvR : R.Reachable u.val v.val := |
| SimpleGraph.ConnectedComponent.exact (u.property.trans v.property.symm) |
| exact huv (Subtype.ext (unique_core (leaves u hu) (leaves v hv) huvR)) |
| · exact False.elim |
| intro x y hxy |
| by_contra hn |
| have h : R.Adj x y := ⟨hxy,hn⟩ |
| rw [hempty] at h |
| exact h |
| end CannonForest |
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| end |
| end |
| end |
| end |
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| end OAI |
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