openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /InvariantFiniteSets.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.AutDisplacement | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OpenGraphIso | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped Classical ENNReal BigOperators | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma vertex_aut_orbit_infinite [Infinite V] (hc : G.fullGraph.Connected) | |
| (hl : G.LocallyFinite) (hq : G.QuasiTransitive) (v : V) : | |
| (Set.range (fun φ : G.Aut => φ.vertex v)).Infinite := by | |
| intro hf | |
| obtain ⟨R,hR⟩ := G.finite_subset_vertexBall hc v hf | |
| obtain ⟨φ,hφ⟩ := G.aut_moves_vertex_outside_ball hc hl hq v R | |
| exact hφ (hR ⟨φ,rfl⟩) | |
| lemma bounded_invariant_marks_null [Infinite V] [Countable V] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) | |
| (p : unitInterval) (M : Set E → V → Prop) | |
| (hm : ∀ v, Measurable (fun ω => M ω v)) | |
| (ha : ∀ (φ : G.Aut) ω v, M (φ.edge '' ω) (φ.vertex v) ↔ M ω v) | |
| (N : ℕ) (hb : ∀ ω, ∃ S : Finset V, S.card≤N ∧ ∀ v, M ω v → v∈S) (v : V) : | |
| G.law p {ω | M ω v}=0 := by | |
| have hmeas (x : V) : Measurable (fun ω => if M ω x then (1:ℝ≥0∞) else 0) := | |
| measurable_const.ite (hm x).setOf measurable_const | |
| have he (x : V) : (∫⁻ ω, if M ω x then (1:ℝ≥0∞) else 0 ∂G.law p) = | |
| G.law p {ω | M ω x} := by | |
| simpa only [Set.indicator,Pi.one_apply,Set.mem_ofPred_eq] using | |
| (lintegral_indicator_one (μ := G.law p) (hm x).setOf) | |
| have hpoint (ω : Set E) : (∑' x, if M ω x then (1:ℝ≥0∞) else 0) ≤ N := by | |
| obtain ⟨S,hS,hcover⟩ := hb ω | |
| calc | |
| _ ≤ ∑' x, if x∈S then (1:ℝ≥0∞) else 0 := by | |
| apply ENNReal.tsum_le_tsum | |
| intro x | |
| split_ifs with hx hs | |
| · rfl | |
| · exact False.elim (hs (hcover x hx)) | |
| · exact zero_le | |
| · rfl | |
| _ = (S.card:ℝ≥0∞) := by | |
| rw [tsum_eq_sum (s := S) (fun x hx => ite_eq_right hx)] | |
| simp | |
| _ ≤ N := by exact_mod_cast hS | |
| have hsum : (∑' x, G.law p {ω | M ω x}) ≤ N := by | |
| calc | |
| _ = ∫⁻ ω, ∑' x, if M ω x then (1:ℝ≥0∞) else 0 ∂G.law p := by | |
| rw [lintegral_tsum (fun x => (hmeas x).aemeasurable)] | |
| simp only [he] | |
| _ ≤ ∫⁻ _ : Set E, (N:ℝ≥0∞) ∂G.law p := lintegral_mono hpoint | |
| _ = N := by simp | |
| have hinv (φ : G.Aut) : G.law p {ω | M ω (φ.vertex v)} = G.law p {ω | M ω v} := by | |
| conv_lhs => rw [←G.law_map_aut p φ] | |
| rw [Measure.map_apply (BernoulliProduct.measurable_image_equiv φ.edge) (hm _).setOf] | |
| congr 1 | |
| ext ω | |
| exact ha φ ω v | |
| by_contra hn | |
| have hf := ENNReal.finite_const_le_of_tsum_ne_top | |
| ((hsum.trans_lt (ENNReal.natCast_lt_top N)).ne) hn | |
| apply G.vertex_aut_orbit_infinite hc hl hq v | |
| apply hf.subset | |
| rintro x ⟨φ,rfl⟩ | |
| change G.law p {ω | M ω v} ≤ G.law p {ω | M ω (φ.vertex v)} | |
| rw [hinv] | |
| def vertexSetBound (_G : BondGraph V E) (A : Set E → Set V) (N : ℕ) (ω : Set E) : Prop := | |
| ∃ S : Finset V, S.card≤N ∧ A ω ⊆ S | |
| lemma measurable_vertexSetBound [Countable V] (A : Set E → Set V) | |
| (hm : ∀ v, Measurable (fun ω => v∈A ω)) (N : ℕ) : | |
| Measurable (G.vertexSetBound A N) := by | |
| exact .exists fun S : Finset V => measurable_const.and | |
| (.forall fun v => (hm v).imp measurable_const) | |
| lemma vertexSetBound_aut (A : Set E → Set V) | |
| (ha : ∀ (φ : G.Aut) ω, A (φ.edge '' ω)=φ.vertex '' A ω) | |
| (N : ℕ) (φ : G.Aut) (ω : Set E) : | |
| G.vertexSetBound A N (φ.edge '' ω) ↔ G.vertexSetBound A N ω := by | |
| change (∃ S : Finset V, S.card≤N ∧ A (φ.edge '' ω)⊆S) ↔ _ | |
| rw [ha] | |
| constructor | |
| · rintro ⟨S,hS,h⟩ | |
| refine ⟨S.image φ.vertex.symm,Finset.card_image_le.trans hS,?_⟩ | |
| intro x hx | |
| exact Finset.mem_image.mpr ⟨φ.vertex x,h ⟨x,hx,rfl⟩,φ.vertex.symm_apply_apply x⟩ | |
| · rintro ⟨S,hS,h⟩ | |
| refine ⟨S.image φ.vertex,Finset.card_image_le.trans hS,?_⟩ | |
| rintro y ⟨x,hx,rfl⟩ | |
| exact Finset.mem_image.mpr ⟨x,h hx,rfl⟩ | |
| theorem invariant_finite_vertex_set_empty_ae [Infinite V] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) | |
| (p : unitInterval) (A : Set E → Set V) | |
| (hm : ∀ v, Measurable (fun ω => v∈A ω)) | |
| (ha : ∀ (φ : G.Aut) ω, A (φ.edge '' ω)=φ.vertex '' A ω) : | |
| ∀ᵐ ω ∂G.law p, (A ω).Finite → A ω=∅ := by | |
| let := G.countable_vertices hc hl | |
| have hn (N : ℕ) (v : V) : ∀ᵐ ω ∂G.law p, ¬(G.vertexSetBound A N ω ∧ v∈A ω) := by | |
| rw [ae_iff] | |
| simp only [not_not] | |
| apply G.bounded_invariant_marks_null hc hl hq p | |
| (fun ω v => G.vertexSetBound A N ω ∧ v∈A ω) | |
| (fun v => (G.measurable_vertexSetBound A hm N).and (hm v)) | |
| (fun φ ω v => by | |
| rw [G.vertexSetBound_aut A ha N φ ω,ha] | |
| simp only [Set.mem_image,φ.vertex.injective.eq_iff,exists_eq_right]) N | |
| intro ω | |
| by_cases hh : G.vertexSetBound A N ω | |
| · obtain ⟨S,hS,hcover⟩ := hh | |
| exact ⟨S,hS,fun x hx => hcover hx.2⟩ | |
| · exact ⟨∅,Nat.zero_le N,fun x hx => (hh hx.1).elim⟩ | |
| have hall : ∀ᵐ ω ∂G.law p, ∀ N : ℕ, ∀ v : V, | |
| ¬(G.vertexSetBound A N ω ∧ v∈A ω) := | |
| ae_all_iff.mpr (fun N => ae_all_iff.mpr (hn N)) | |
| filter_upwards [hall] with ω hω hf | |
| apply Set.eq_empty_iff_forall_notMem.mpr | |
| intro v hv | |
| exact hω hf.toFinset.card v ⟨⟨hf.toFinset,le_rfl,fun _ hx => hf.mem_toFinset.mpr hx⟩,hv⟩ | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |