openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Modular /ExpectedIncidenceTransport.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.Modular.NormalizedIncidenceTransport | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory Filter | |
| open scoped Classical ENNReal BigOperators | |
| namespace CriticalPercolation | |
| lemma lintegral_ne_top_of_uniform_bound {α : Type*} [MeasurableSpace α] | |
| (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) (C : ℝ≥0∞) (hC : C≠∞) | |
| (hb : ∀ a, f a≤C) : (∫⁻ a, f a ∂μ)≠∞ := by | |
| apply ne_top_of_le_ne_top (ENNReal.mul_ne_top hC (measure_ne_top μ Set.univ)) | |
| simpa only [lintegral_const] using lintegral_mono hb | |
| lemma sum_integral_toReal {α ι : Type*} [MeasurableSpace α] (μ : Measure α) [IsFiniteMeasure μ] | |
| (S : Finset ι) (F : ι → α → ℝ≥0∞) (hm : ∀ i∈S, Measurable (F i)) | |
| (C : ι → ℝ≥0∞) (hC : ∀ i∈S, C i≠∞) (hb : ∀ i∈S, ∀ a, F i a≤C i) : | |
| (∑ i∈S, ∫ a, (F i a).toReal ∂μ)=(∑ i∈S, ∫⁻ a, F i a ∂μ).toReal := by | |
| rw [ENNReal.toReal_sum] | |
| · apply Finset.sum_congr rfl | |
| intro i hi | |
| exact integral_toReal (hm i hi).aemeasurable (Eventually.of_forall (fun a => | |
| lt_top_iff_ne_top.mpr (ne_top_of_le_ne_top (hC i hi) (hb i hi a)))) | |
| · intro i hi | |
| exact lintegral_ne_top_of_uniform_bound μ (F i) (C i) (hC i hi) (hb i hi) | |
| namespace BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| def selectedIncidenceReturn (hl : G.LocallyFinite) (H : Set E → E → ℝ≥0∞) | |
| (ω : Set E) (y : V) : ℝ≥0∞ := | |
| ∑ e∈(hl y).toFinset, H ω e*G.incidentClusterCount ω e | |
| lemma selectedIncidenceReturn_eq_tsum (hl : G.LocallyFinite) (H : Set E → E → ℝ≥0∞) | |
| (ω : Set E) (y : V) : G.selectedIncidenceReturn hl H ω y = | |
| ∑' e, if y∈G.ends e then H ω e*G.incidentClusterCount ω e else 0 := by | |
| rw [tsum_eq_sum (s := (hl y).toFinset) (fun e he => ite_eq_right (by simpa using he))] | |
| apply Finset.sum_congr rfl | |
| intro e he | |
| exact (ite_eq_left ((hl y).mem_toFinset.mp he)).symm | |
| lemma measurable_selectedIncidenceReturn [Countable V] [Countable E] | |
| (hl : G.LocallyFinite) (H : Set E → E → ℝ≥0∞) (hm : ∀ e, Measurable (fun ω => H ω e)) (y : V) : | |
| Measurable (fun ω => G.selectedIncidenceReturn hl H ω y) := | |
| Finset.measurable_sum _ (fun e _ => (hm e).mul (G.measurable_incidentClusterCount e)) | |
| lemma selectedIncidenceReturn_le (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (H : Set E → E → ℝ≥0∞) (hH : ∀ ω e, H ω e≤1) (ω : Set E) (y : V) : | |
| G.selectedIncidenceReturn hl H ω y≤(hl y).toFinset.card*2 := by | |
| unfold selectedIncidenceReturn | |
| calc _ ≤ ∑ _e∈(hl y).toFinset, (2:ℝ≥0∞) := by | |
| apply Finset.sum_le_sum | |
| intro e _ | |
| exact (mul_le_mul' (hH ω e) (G.incidentClusterCount_le_two hc hl ω e)).trans_eq (one_mul 2) | |
| _ = _ := by simp | |
| lemma selectedIncidenceReturn_toReal (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (H : Set E → E → ℝ≥0∞) (hH : ∀ ω e, H ω e≤1) (ω : Set E) (y : V) : | |
| (G.selectedIncidenceReturn hl H ω y).toReal= | |
| ∑ e∈(hl y).toFinset, (H ω e).toReal*(G.incidentClusterCount ω e).toReal := by | |
| unfold selectedIncidenceReturn | |
| rw [ENNReal.toReal_sum] | |
| · simp only [ENNReal.toReal_mul] | |
| · intro e _ | |
| exact ENNReal.mul_ne_top (ne_top_of_le_ne_top ENNReal.one_ne_top (hH ω e)) | |
| (ne_top_of_le_ne_top (by norm_num : (2:ℝ≥0∞)≠∞) (G.incidentClusterCount_le_two hc hl ω e)) | |
| lemma expected_selectedIncidence_real [Countable V] [Countable E] | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) | |
| (D K : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D) | |
| (hK : ∀ v, (G.vertexBall v 1).ncard≤K) | |
| (p : unitInterval) (H : Set E → E → ℝ≥0∞) | |
| (hm : ∀ e, Measurable (fun ω => H ω e)) (hH : ∀ ω e, H ω e≤1) | |
| (ha : ∀ (φ : G.Aut) ω e, H (φ.edge '' ω) (φ.edge e)=H ω e) : | |
| (∑ x∈(G.orbitRepresentatives hq).roots, ∫ ω, (G.normalizedSelectedIncidence H ω x).toReal ∂G.law p)= | |
| ∑ y∈(G.orbitRepresentatives hq).roots, ∫ ω, (G.selectedIncidenceReturn hl H ω y).toReal ∂G.law p := by | |
| rw [sum_integral_toReal (G.law p) _ _ (fun x _ => G.measurable_normalizedSelectedIncidence H hm x) | |
| (fun _ => (D*(2*K) : ℕ)) (fun _ _ => ENNReal.natCast_ne_top _) | |
| (fun x _ ω => G.normalizedSelectedIncidence_le hc hl D K hD hK H hH ω x), | |
| sum_integral_toReal (G.law p) _ _ (fun y _ => G.measurable_selectedIncidenceReturn hl H hm y) | |
| (fun y => (hl y).toFinset.card*2) (fun _ _ => ENNReal.mul_ne_top (ENNReal.natCast_ne_top _) (by norm_num)) | |
| (fun y _ ω => G.selectedIncidenceReturn_le hc hl H hH ω y)] | |
| congr 1 | |
| simp_rw [G.selectedIncidenceReturn_eq_tsum] | |
| exact G.expected_normalizedSelectedIncidence hc hl hq p H hm ha | |
| end BondGraph | |
| end CriticalPercolation | |
| end | |
| end | |
| end OAI | |