| import Mathlib |
|
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| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonForest |
| open Set |
|
|
| theorem empty_of_no_leaves {V : Type*} [Finite V] (G : SimpleGraph V) |
| (hG : G.IsAcyclic) |
| (hother : ∀ x y,G.Adj x y → ∃ z,G.Adj x z ∧ z≠y) : G=⊥ := by |
| classical |
| ext x y |
| constructor |
| · intro hxy |
| let c := G.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 ⟨v,hv⟩ := (hG.isTree_connectedComponent c).exists_vert_degree_one_of_nontrivial |
| obtain ⟨w,hw,hunique⟩ := SimpleGraph.degree_eq_one_iff_existsUnique_adj.mp hv |
| have hval : G.Adj v.val w.val := hw |
| obtain ⟨z,hz,hzw⟩ := hother v.val w.val hval |
| have hzc : z∈c.supp := c.mem_supp_of_adj_mem_supp v.property hz |
| let z' : c.supp := ⟨z,hzc⟩ |
| have hvz : c.toSimpleGraph.Adj v z' := hz |
| exact (hzw (congrArg Subtype.val (hunique _ hvz))).elim |
| · exact False.elim |
| end CannonForest |
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|
| end |
| end |
| section |
| noncomputable section |
| namespace CannonGraphGauge |
| open Set |
|
|
| variable {V E D : Type*} [Fintype E] [Group D] |
| def edgeGraph (a b : E→V) (hneq:∀i,a i≠b i) (S:Set E) : SimpleGraph V where |
| Adj x y := ∃ i∈S,(a i=x ∧ b i=y) ∨ (a i=y ∧ b i=x) |
| symm := ⟨by |
| rintro x y ⟨i,hi,h|h⟩ |
| · exact ⟨i,hi,Or.inr h⟩ |
| · exact ⟨i,hi,Or.inl h⟩⟩ |
| loopless := ⟨by rintro x ⟨i,hi,h|h⟩ <;> exact hneq i (h.1.trans h.2.symm)⟩ |
| |
| def goodEdges (a b:E→V) (K:E→D) (v:V→D) : Finset E := by |
| classical |
| exact Finset.univ.filter fun i=>K i*v (a i)=v (b i) |
| @[simp] lemma mem_goodEdges (a b:E→V) (K:E→D) (v:V→D) (i:E) : |
| i∈goodEdges a b K v ↔ K i*v (a i)=v (b i) := by |
| classical |
| simp [goodEdges] |
|
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| theorem exists_connected_good (a b:E→V) (hneq:∀i,a i≠b i) (K:E→D) |
| (hconn:(edgeGraph a b hneq univ).Preconnected) : |
| ∃ v:V→D,(edgeGraph a b hneq (goodEdges a b K v : Set E)).Preconnected := by |
| classical |
| let S : Finset (Finset E) := Finset.univ.filter fun A=>∃ v:V→D,goodEdges a b K v=A |
| have hS : S.Nonempty := ⟨goodEdges a b K (fun _=>1),by simp [S]⟩ |
| obtain ⟨A,hAS,hmax⟩ := Finset.exists_max_image S Finset.card hS |
| obtain ⟨v,rfl⟩ := (Finset.mem_filter.mp hAS).2 |
| have hmaxv (w:V→D) : (goodEdges a b K w).card≤(goodEdges a b K v).card := |
| hmax _ (by simp [S]) |
| let H := edgeGraph a b hneq (goodEdges a b K v : Set E) |
| have hreach (i:E) : H.Reachable (a i) (b i) := by |
| by_contra hn |
| let C (x:V) : Prop := H.Reachable (a i) x |
| let γ := (K i*v (a i))⁻¹*v (b i) |
| let w (x:V) : D := if C x then v x*γ else v x |
| have hpres : goodEdges a b K v⊆goodEdges a b K w := by |
| intro j hj |
| have hadj : H.Adj (a j) (b j) := ⟨j,hj,Or.inl ⟨rfl,rfl⟩⟩ |
| have hc : C (a j)↔C (b j) := |
| ⟨fun h=>h.trans hadj.reachable,fun h=>h.trans hadj.reachable.symm⟩ |
| apply (mem_goodEdges ..).mpr |
| have he := (mem_goodEdges a b K v j).mp hj |
| by_cases ha : C (a j) |
| · simp only [w,ite_eq_left ha,ite_eq_left (hc.mp ha),←mul_assoc,he] |
| · simp only [w,ite_eq_right ha,ite_eq_right (mt hc.mpr ha),he] |
| have hiw : i∈goodEdges a b K w := by |
| apply (mem_goodEdges ..).mpr |
| have hai : C (a i) := .rfl |
| have hbi : ¬C (b i) := hn |
| simp [w,hai,hbi,γ,←mul_assoc] |
| have hiv : i∉goodEdges a b K v := by |
| intro hi |
| exact hn (show H.Adj (a i) (b i) from ⟨i,hi,Or.inl ⟨rfl,rfl⟩⟩).reachable |
| have hlt := Finset.card_lt_card (Finset.ssubset_iff_subset_ne.mpr |
| ⟨hpres,fun he=>hiv (he ▸ hiw)⟩) |
| exact (not_lt_of_ge (hmaxv w)) hlt |
| refine ⟨v,?_⟩ |
| intro x y |
| obtain ⟨p⟩ := hconn x y |
| induction p with |
| | nil => exact .rfl |
| | @cons x y z h p ih => |
| obtain ⟨i,-,h|h⟩ := h |
| · exact (h.1 ▸ h.2 ▸ hreach i).trans ih |
| · exact (h.2 ▸ h.1 ▸ (hreach i).symm).trans ih |
|
|
| theorem exists_connected_good_restrict (a b:E→V) (hneq:∀i,a i≠b i) (K:E→D) |
| (R:Set E) (hconn:(edgeGraph a b hneq R).Preconnected) : |
| ∃ v:V→D,(edgeGraph a b hneq (R∩(goodEdges a b K v : Set E))).Preconnected := by |
| classical |
| let I := {i // i∈R} |
| let _ : Fintype I := Fintype.ofFinite I |
| let a' : I→V := fun i=>a i.val |
| let b' : I→V := fun i=>b i.val |
| have hab' : ∀ i:I,a' i≠b' i := fun i=>hneq i.val |
| have heq0 : edgeGraph a' b' hab' univ=edgeGraph a b hneq R := by |
| ext x y |
| constructor |
| · rintro ⟨i,-,h⟩; exact ⟨i.val,i.property,h⟩ |
| · rintro ⟨i,hi,h⟩; exact ⟨⟨i,hi⟩,mem_univ _,h⟩ |
| obtain ⟨v,hv⟩ := exists_connected_good a' b' hab' (fun i=>K i.val) (heq0 ▸ hconn) |
| refine ⟨v,?_⟩ |
| have heq : edgeGraph a' b' hab' (goodEdges a' b' (fun i=>K i.val) v : Set I)= |
| edgeGraph a b hneq (R∩(goodEdges a b K v : Set E)) := by |
| ext x y |
| constructor |
| · rintro ⟨i,hi,h⟩ |
| exact ⟨i.val,⟨i.property,by simpa [a',b'] using hi⟩,h⟩ |
| · rintro ⟨i,⟨hi,hg⟩,h⟩ |
| exact ⟨⟨i,hi⟩,by simpa [a',b'] using hg,h⟩ |
| rwa [heq] at hv |
|
|
| theorem label_equal_of_other_edges {W Z : Type*} [Finite W] (G:SimpleGraph W) |
| (hG:G.IsCycles) {a b:W} (hab:G.Adj a b) (f:W→Z) |
| (hother:∀x y,G.Adj x y → s(x,y)≠s(a,b) → f x=f y) : f a=f b := by |
| obtain ⟨p⟩ := hG.reachable_deleteEdges hab |
| have hw {x y:W} (p:(G.deleteEdges {s(a,b)}).Walk x y) : f x=f y := by |
| induction p with |
| | nil => rfl |
| | @cons x y z h p ih => |
| have hx : G.Adj x y ∧ s(x,y)∉({s(a,b)}:Set (Sym2 W)) := SimpleGraph.deleteEdges_adj.mp h |
| exact (hother x y hx.1 (by simpa using hx.2)).trans ih |
| exact hw p |
| end CannonGraphGauge |
|
|
| end |
| end |
| end |
| end |
|
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| end OAI |
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