openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /ProductMeasure /FiniteAdjacentMeasure.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.BitMeasure | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteAdjacent | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced | |
| import OAI.Probability.CriticalPercolation.Modular.FiniteClusterTransport | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory | |
| open scoped Classical | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma finite_averaged_adjacent_measure [Fintype V] [Fintype E] | |
| (p : unitInterval) (hp : (p : ℝ) ∈ Set.Ioo 0 1) | |
| (D : ℕ) (hDpos : 1≤D) (hD : ∀ v, (G.incidentEdges {v}).ncard ≤ D) | |
| (x y : E → V) (he : ∀ e, G.ends e = s(x e,y e)) (K : ℕ) : | |
| (∑ e, (G.law p).real (G.twoClusterEvent (x e) (y e) (4^K))) ≤ | |
| (Fintype.card V : ℝ)*((D:ℝ)/(p:ℝ)*(6/(2:ℝ)^K)) := by | |
| have h := G.finite_averaged_adjacent p hp (fun _ : E => BernoulliProduct.bitWeight p) | |
| (fun _ => BernoulliProduct.bitWeight_sum p) (fun _ => BernoulliProduct.bitWeight_nonneg p) | |
| (fun _ => by simp [BernoulliProduct.bitWeight]) | |
| (fun _ => by simp [BernoulliProduct.bitWeight]) D hDpos | |
| (fun v => by simpa only [← Set.ncard_eq_toFinset_card'] using hD v) x y he K | |
| simpa only [G.finite_endpoint_prob] using h | |
| lemma induce_incident_image_subset (S : Set V) (v : S) : | |
| Subtype.val '' ((G.induce S).incidentEdges {v}) ⊆ G.incidentEdges {v.val} := by | |
| rintro e ⟨a,ha,rfl⟩ | |
| simp only [incidentEdges,Set.mem_ofPred_eq,Set.mem_singleton_iff,exists_eq_left] at ha ⊢ | |
| rw [← G.induce_ends_map S a] | |
| exact Sym2.mem_map.mpr ⟨v,ha,rfl⟩ | |
| lemma induce_incident_bound (hlocal : G.LocallyFinite) (D : ℕ) | |
| (hD : ∀ v, (G.incidentEdges {v}).ncard ≤ D) (S : Set V) (v : S) : | |
| ((G.induce S).incidentEdges {v}).ncard ≤ D := by | |
| calc | |
| _ = (Subtype.val '' ((G.induce S).incidentEdges {v})).ncard := | |
| (Set.ncard_image_of_injective _ Subtype.val_injective).symm | |
| _ ≤ (G.incidentEdges {v.val}).ncard := Set.ncard_le_ncard | |
| (G.induce_incident_image_subset S v) (G.finite_incident hlocal (Set.finite_singleton _)) | |
| _ ≤ D := hD v.val | |
| lemma induced_averaged_adjacent (hlocal : G.LocallyFinite) | |
| (p : unitInterval) (hp : (p : ℝ) ∈ Set.Ioo 0 1) | |
| (D : ℕ) (hDpos : 1≤D) (hD : ∀ v, (G.incidentEdges {v}).ncard ≤ D) | |
| (S : Set V) (hS : S.Finite) (K : ℕ) : | |
| ∑ e ∈ (G.finite_inducedEdges hlocal hS).toFinset, | |
| (G.law p).real {ω | G.inducedEdges S ∩ ω ∈ | |
| G.twoClusterEvent (G.ends e).out.1 (G.ends e).out.2 (4^K)} ≤ | |
| (S.ncard : ℝ)*((D:ℝ)/(p:ℝ)*(6/(2:ℝ)^K)) := by | |
| let := hS.fintype | |
| let := (G.finite_inducedEdges hlocal hS).fintype | |
| let x (e : G.inducedEdges S) : S := ⟨(G.ends e.val).out.1,e.property _ (Sym2.out_fst_mem _)⟩ | |
| let y (e : G.inducedEdges S) : S := ⟨(G.ends e.val).out.2,e.property _ (Sym2.out_snd_mem _)⟩ | |
| have he (e : G.inducedEdges S) : (G.induce S).ends e = s(x e,y e) := by | |
| apply (G.induce_ends_eq S e (x e) (y e)).mpr | |
| exact (G.ends e.val).out_eq.symm | |
| have hh := (G.induce S).finite_averaged_adjacent_measure p hp D hDpos | |
| (G.induce_incident_bound hlocal D hD S) x y he K | |
| simp_rw [G.induce_twoCluster_real S (G.finite_inducedEdges hlocal hS)] at hh | |
| rw [Finset.sum_subtype (G.finite_inducedEdges hlocal hS).toFinset | |
| (fun e => (G.finite_inducedEdges hlocal hS).mem_toFinset)] | |
| simpa only [x,y,Set.ncard_eq_toFinset_card',Set.toFinset_card] using hh | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |