openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /GraphGeometry /FiniteSumConnection.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.Leak | |
| import OAI.Probability.CriticalPercolation.Criticality.SharpnessPhi | |
| namespace OAI | |
| section | |
| section CurrentModule7 | |
| noncomputable section | |
| open Set MeasureTheory | |
| open scoped Classical BigOperators | |
| namespace CriticalPercolation.Sharpness | |
| open ClusterDeletion | |
| variable {V E : Type*} (G : BondGraph V E) | |
| lemma finite_sum_connection_le [Fintype E] {p : ℝ} (hp : p ∈ Icc 0 1) | |
| (A : Finset V) (S : Set V) (hS : S.Finite) (o : V) (ho : o ∈ S) | |
| (M : ℝ) (hM : ∀ y : V, ∑ v ∈ A, | |
| probability p (fun ω => (G.openGraph ω).Reachable y v) ≤ M) : | |
| ∑ v ∈ A, probability p (fun ω => (G.openGraph ω).Reachable o v) ≤ | |
| S.ncard + boundaryFlux G p o Sᶜ * M := by | |
| let f := fun x y => probability p (fun ω => (G.openGraph ω).Reachable x y) | |
| let c := fun e => p * probability p (fun ω => | |
| boundaryReach G Sᶜ (delete (G.openGraph ω) Sᶜ) o e) | |
| have hc : ∀ e, 0 ≤ c e := fun e => mul_nonneg hp.1 (probability_nonneg hp _) | |
| have hout (v) (hv : v ∉ S) : f o v ≤ ∑ e, c e * f (cutOutside G S e) v := | |
| simon_lieb G hp S o ho v hv | |
| calc | |
| _ ≤ ∑ v ∈ A, ((if v ∈ S then (1:ℝ) else 0) + | |
| ∑ e, c e * f (cutOutside G S e) v) := by | |
| apply Finset.sum_le_sum | |
| intro v _ | |
| by_cases hv : v ∈ S | |
| · simp only [ite_eq_left hv] | |
| exact (probability_le_one hp _).trans (le_add_of_nonneg_right | |
| (Finset.sum_nonneg fun e _ => mul_nonneg (hc e) (probability_nonneg hp _))) | |
| · simpa only [ite_eq_right hv,zero_add] using hout v hv | |
| _ = (A.filter (· ∈ S)).card + ∑ e, c e * ∑ v ∈ A, f (cutOutside G S e) v := by | |
| rw [Finset.sum_add_distrib,Finset.sum_comm] | |
| simp only [← Finset.mul_sum] | |
| congr 1 | |
| simp | |
| _ ≤ S.ncard + (∑ e, c e) * M := by | |
| apply add_le_add | |
| · exact_mod_cast Set.ncard_le_ncard | |
| (show (↑(A.filter (· ∈ S)) : Set V) ⊆ S from fun v hv => (Finset.mem_filter.mp hv).2) hS | |
| · rw [Finset.sum_mul] | |
| exact Finset.sum_le_sum fun e _ => mul_le_mul_of_nonneg_left (hM _) (hc e) | |
| _ = _ := by | |
| rw [boundaryFlux_eq_sum,Finset.mul_sum] | |
| lemma restricted_boundaryFlux_le_phi (hl : G.LocallyFinite) (p : unitInterval) | |
| (S : Set V) (hS : S.Finite) (o : V) (K : Set E) (hK : K.Finite) : | |
| letI := hK.fintype | |
| boundaryFlux (G.restrictEdges K) (p : ℝ) o Sᶜ ≤ phi G p o S := by | |
| let := hK.fintype | |
| let F := fun e => (G.law p).real {ω | boundaryReach G Sᶜ | |
| (delete (G.openGraph ω) Sᶜ) o e} | |
| have hF : Summable F := summable_of_hasFiniteSupport | |
| ((finite_incident G hl hS).subset (boundary_measure_support G p o S)) | |
| rw [boundaryFlux_eq_sum,phi] | |
| apply mul_le_mul_of_nonneg_left _ p.property.1 | |
| calc | |
| _ ≤ ∑ e : K, F e.val := by | |
| apply Finset.sum_le_sum | |
| intro e _ | |
| rw [finite_probability_restrict G p K hK] | |
| apply measureReal_mono _ (measure_ne_top _ _) | |
| intro ω hω | |
| change boundaryReach (G.restrictEdges K) Sᶜ | |
| (delete ((G.restrictEdges K).openGraph (Subtype.val ⁻¹' ω)) Sᶜ) o e at hω | |
| rw [G.restrictEdges_openGraph] at hω | |
| have hh : boundaryReach G Sᶜ (delete (G.openGraph (K ∩ ω)) Sᶜ) o e.val := hω | |
| obtain ⟨x,y,he,hx,hy,hr⟩ := hh | |
| have hdel : delete (G.openGraph (K ∩ ω)) Sᶜ ≤ delete (G.openGraph ω) Sᶜ := | |
| fun _ _ hab => ⟨G.openGraph_mono inter_subset_right hab.1,hab.2⟩ | |
| exact ⟨x,y,he,hx,hy,hr.mono hdel⟩ | |
| _ ≤ ∑' e, F e := by | |
| simpa only [tsum_fintype,Function.comp_def] using | |
| (tsum_comp_le_tsum_of_inj hF (fun e => measureReal_nonneg) | |
| (Subtype.val_injective : Function.Injective (Subtype.val : K → E))) | |
| end CriticalPercolation.Sharpness | |
| end | |
| end CurrentModule7 | |
| end | |
| end OAI | |