| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OpenGraphIso |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set |
| open scoped Classical |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma vertexBall_mono (o : V) : Monotone (G.vertexBall o) := by |
| apply monotone_nat_of_le_succ |
| intro n |
| exact Set.subset_union_left |
|
|
| lemma mem_vertexBall_self (o : V) (n : ℕ) : o ∈ G.vertexBall o n := |
| G.vertexBall_mono o (Nat.zero_le n) rfl |
|
|
| lemma vertexBall_step {o v w : V} {n : ℕ} (hv : v ∈ G.vertexBall o n) |
| (he : G.fullGraph.Adj v w) : w ∈ G.vertexBall o (n+1) := |
| Or.inr (Set.mem_iUnion₂.mpr ⟨v,hv,he⟩) |
|
|
| lemma vertexBall_add {o v : V} {n : ℕ} (hv : v ∈ G.vertexBall o n) (m : ℕ) : |
| G.vertexBall v m ⊆ G.vertexBall o (n+m) := by |
| induction m with |
| | zero => simpa only [vertexBall,Nat.add_zero,Set.singleton_subset_iff] using hv |
| | succ m ih => |
| rintro w (hw | hw) |
| · exact G.vertexBall_mono o (by omega) (ih hw) |
| · obtain ⟨z,hz,hzw⟩ := Set.mem_iUnion₂.mp hw |
| exact G.vertexBall_step (ih hz) hzw |
|
|
| lemma vertexBall_comm {o v : V} {n : ℕ} : v ∈ G.vertexBall o n ↔ o ∈ G.vertexBall v n := by |
| have h : ∀ n (o v : V), v ∈ G.vertexBall o n → o ∈ G.vertexBall v n := by |
| intro n |
| induction n with |
| | zero => intro o v hv; simpa only [vertexBall,Set.mem_singleton_iff] using hv.symm |
| | succ n ih => |
| intro o v hv |
| rcases hv with hv | hv |
| · exact G.vertexBall_mono v (Nat.le_succ n) (ih o v hv) |
| · obtain ⟨z,hz,hzv⟩ := Set.mem_iUnion₂.mp hv |
| have h1 : z ∈ G.vertexBall v 1 := G.vertexBall_step (G.mem_vertexBall_self v 0) hzv.symm |
| simpa only [Nat.add_comm 1] using G.vertexBall_add h1 n (ih o z hz) |
| exact ⟨h n o v,h n v o⟩ |
|
|
| lemma Aut.fullGraph_adj (φ : G.Aut) (u v : V) : |
| G.fullGraph.Adj (φ.vertex u) (φ.vertex v) ↔ G.fullGraph.Adj u v := by |
| simpa only [fullGraph,Set.image_univ,Equiv.range_eq_univ] using Aut.adj_map G φ Set.univ u v |
|
|
| lemma Aut.vertexBall_mem (φ : G.Aut) (o v : V) (n : ℕ) : |
| φ.vertex v ∈ G.vertexBall (φ.vertex o) n ↔ v ∈ G.vertexBall o n := by |
| induction n generalizing v with |
| | zero => simp only [vertexBall,Set.mem_singleton_iff,φ.vertex.injective.eq_iff] |
| | succ n ih => |
| simp only [vertexBall,Set.mem_union,Set.mem_iUnion,exists_prop,SimpleGraph.mem_neighborSet] |
| constructor |
| · rintro (h | ⟨z,hz,hzv⟩) |
| · exact Or.inl ((ih v).mp h) |
| · obtain ⟨w,rfl⟩ := φ.vertex.surjective z |
| exact Or.inr ⟨w,(ih w).mp hz, |
| (Aut.fullGraph_adj G φ w v).mp hzv⟩ |
| · rintro (h | ⟨z,hz,hzv⟩) |
| · exact Or.inl ((ih v).mpr h) |
| · exact Or.inr ⟨φ.vertex z,(ih z).mpr hz,(Aut.fullGraph_adj G φ z v).mpr hzv⟩ |
|
|
| lemma Aut.vertexBall_image (φ : G.Aut) (o : V) (n : ℕ) : |
| φ.vertex '' G.vertexBall o n = G.vertexBall (φ.vertex o) n := by |
| ext v |
| obtain ⟨w,rfl⟩ := φ.vertex.surjective v |
| simpa only [Set.mem_image,φ.vertex.injective.eq_iff,exists_eq_right] using |
| (Aut.vertexBall_mem G φ o w n).symm |
|
|
| lemma Aut.vertexBall_ncard (φ : G.Aut) (o : V) (n : ℕ) : |
| (G.vertexBall (φ.vertex o) n).ncard = (G.vertexBall o n).ncard := by |
| rw [← Aut.vertexBall_image G φ,Set.ncard_image_of_injective _ φ.vertex.injective] |
|
|
| lemma Aut.incident_singleton_image (φ : G.Aut) (v : V) : |
| φ.edge '' G.incidentEdges {v} = G.incidentEdges {φ.vertex v} := by |
| ext e |
| obtain ⟨f,rfl⟩ := φ.edge.surjective e |
| simp only [Set.mem_image,φ.edge.injective.eq_iff,exists_eq_right,incidentEdges, |
| Set.mem_ofPred_eq,Set.mem_singleton_iff,exists_eq_left,φ.preserves] |
| simp only [Sym2.mem_map,φ.vertex.injective.eq_iff,exists_eq_right] |
|
|
| lemma uniform_incident_bound (hquasi : G.QuasiTransitive) : |
| ∃ D : ℕ, 1≤D ∧ ∀ v, (G.incidentEdges {v}).ncard ≤ D := by |
| obtain ⟨S,hS,hcover⟩ := hquasi |
| let D := 1 + ∑ v ∈ hS.toFinset, (G.incidentEdges {v}).ncard |
| refine ⟨D,by dsimp [D]; omega,?_⟩ |
| intro v |
| obtain ⟨z,hz,φ,rfl⟩ := hcover v |
| rw [← Aut.incident_singleton_image G φ,Set.ncard_image_of_injective _ φ.edge.injective] |
| exact (Finset.single_le_sum (f := fun v => (G.incidentEdges {v}).ncard) (fun _ _ => Nat.zero_le _) (hS.mem_toFinset.mpr hz)).trans (Nat.le_add_left _ _) |
|
|
| lemma uniform_ball_bound (hquasi : G.QuasiTransitive) (n : ℕ) : |
| ∃ M : ℕ, ∀ v, (G.vertexBall v n).ncard ≤ M := by |
| obtain ⟨S,hS,hcover⟩ := hquasi |
| refine ⟨∑ v ∈ hS.toFinset, (G.vertexBall v n).ncard,?_⟩ |
| intro v |
| obtain ⟨z,hz,φ,rfl⟩ := hcover v |
| rw [Aut.vertexBall_ncard G φ] |
| exact Finset.single_le_sum (f := fun v => (G.vertexBall v n).ncard) (fun _ _ => Nat.zero_le _) (hS.mem_toFinset.mpr hz) |
|
|
| lemma orbit_density_radius (hconnected : G.fullGraph.Connected) |
| (hquasi : G.QuasiTransitive) (u : V) : |
| ∃ C : ℕ, ∀ v, ∃ φ : G.Aut, φ.vertex u ∈ G.vertexBall v C := by |
| obtain ⟨S,hS,hcover⟩ := hquasi |
| have hpath : ∀ z : S, ∃ n, u ∈ G.vertexBall z.val n := |
| fun z => G.reachable_mem_vertexBall (hconnected z u) |
| choose n hn using hpath |
| let := hS.fintype |
| let C := ∑ z : S, n z |
| refine ⟨C,?_⟩ |
| intro v |
| obtain ⟨z,hz,φ,rfl⟩ := hcover v |
| refine ⟨φ,(Aut.vertexBall_mem G φ z u C).mpr ?_⟩ |
| exact G.vertexBall_mono z (Finset.single_le_sum (fun _ _ => Nat.zero_le _) |
| (Finset.mem_univ (⟨z,hz⟩ : S))) (hn ⟨z,hz⟩) |
|
|
| lemma finite_subset_vertexBall (hconnected : G.fullGraph.Connected) |
| (o : V) {S : Set V} (hS : S.Finite) : ∃ n, S ⊆ G.vertexBall o n := by |
| have hh : ∀ v : S, ∃ n, v.val ∈ G.vertexBall o n := |
| fun v => G.reachable_mem_vertexBall (hconnected o v) |
| choose n hn using hh |
| let := hS.fintype |
| refine ⟨∑ v : S, n v,?_⟩ |
| intro v hv |
| exact G.vertexBall_mono o (Finset.single_le_sum (fun _ _ => Nat.zero_le _) |
| (Finset.mem_univ (⟨v,hv⟩ : S))) (hn ⟨v,hv⟩) |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
|
|
