openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Modular /MarkedTiltedTransport.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Modular.TiltedTransportENNReal | |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneBernoulli | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped Classical ENNReal BigOperators | |
| namespace CriticalPercolation.BondGraph | |
| open BernoulliProduct | |
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) | |
| omit [Countable V] [Countable E] in | |
| lemma expected_mark_transport_aut | |
| (F : (E → ℝ) → V → V → ℝ≥0∞) (hm : ∀ x y, Measurable (fun u => F u x y)) | |
| (ha : ∀ (φ : G.Aut) u x y, | |
| F (reindexMarks φ.edge u) (φ.vertex x) (φ.vertex y) = F u x y) | |
| (φ : G.Aut) (x y : V) : | |
| (∫⁻ u, F u (φ.vertex x) (φ.vertex y) ∂markLaw) = ∫⁻ u, F u x y ∂markLaw := by | |
| conv_lhs => rw [←markLaw_map_reindex φ.edge] | |
| rw [lintegral_map (hm _ _) (measurable_reindexMarks φ.edge)] | |
| simp only [ha] | |
| omit [Countable E] in | |
| lemma OrbitRepresentatives.mark_tilted_transport (R : G.OrbitRepresentatives) | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (F : (E → ℝ) → V → V → ℝ≥0∞) (hm : ∀ x y, Measurable (fun u => F u x y)) | |
| (ha : ∀ (φ : G.Aut) u x y, | |
| F (reindexMarks φ.edge u) (φ.vertex x) (φ.vertex y) = F u x y) : | |
| (∑ x ∈ R.roots, ∫⁻ u, ∑' y, F u x y ∂markLaw) = | |
| ∑ y ∈ R.roots, ∫⁻ u, ∑' x, F u x y * G.stabilizerWeight y x ∂markLaw := by | |
| simp_rw [lintegral_tsum (fun y => (hm _ y).aemeasurable), | |
| lintegral_tsum (fun x => ((hm x _).mul_const _).aemeasurable), | |
| lintegral_mul_const _ (hm _ _)] | |
| exact OrbitRepresentatives.tilted_massTransport_ennreal G R hc hl _ | |
| (G.expected_mark_transport_aut F hm ha) | |
| omit [Countable V] [Countable E] in | |
| lemma marks_ae_of_law_ae (p : unitInterval) (P : Set E → Prop) (h : ∀ᵐ ω ∂G.law p, P ω) : | |
| ∀ᵐ u ∂markLaw, P (atParameter p u) := by | |
| apply ae_of_ae_map (measurable_atParameter p).aemeasurable | |
| rwa [markLaw_map_atParameter,BernoulliProduct.law_const] | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |