openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /InfiniteInterpolation.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.Criticality.InterpolationUniform | |
| import OAI.Probability.CriticalPercolation.Criticality.ExhaustionEvents | |
| import OAI.Probability.CriticalPercolation.Criticality.RestrictedInterpolation | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical Topology | |
| namespace CriticalPercolation | |
| lemma exists_le_of_eventually_exists_le {I : Type*} [Finite I] | |
| {f : I → ℕ → ℝ} {a : I → ℝ} {r : ℕ → ℝ} {b : ℝ} | |
| (hf : ∀ i, Tendsto (f i) atTop (𝓝 (a i))) (hr : Tendsto r atTop (𝓝 b)) | |
| (h : ∀ᶠ n in atTop, ∃ i, r n ≤ f i n) : ∃ i, b ≤ a i := by | |
| by_contra! hh | |
| have ht : ∀ᶠ n in atTop, ∀ i, f i n < r n := | |
| Filter.eventually_all.mpr (fun i => (hf i).eventually_lt hr (hh i)) | |
| obtain ⟨n,hn,hn'⟩ := (h.and ht).exists | |
| obtain ⟨i,hi⟩ := hn | |
| exact (not_le.mpr (hn' i)) hi | |
| namespace BondGraph | |
| variable {V E : Type*} [Countable E] (G : BondGraph V E) | |
| theorem infinite_interpolation_measure | |
| (d : ℕ) (x : ℕ → V) (hd : 2≤d) (p : unitInterval) | |
| (hp : (p : ℝ) ∈ Set.Ioo 0 1) {c ε : ℝ} (hc : 0<c) (hε : 0<ε) | |
| (hεp : ε≤p) (hεq : ε≤1-(p:ℝ)) | |
| (hconn : ∀ j k, j<k → k≤d → k-j<d → | |
| c ≤ (G.law p).real {ω | (G.openGraph ω).Reachable (x j) (x k)}) | |
| (s : ℕ) (hδ : 0 < (G.law p).real (G.twoClusterEvent (x 0) (x d) s)) : | |
| ∃ j<d, 0 < (G.law p).real (G.twoClusterEvent (x j) (x (j+1)) s) ∧ | |
| -Real.log ((G.law p).real (G.twoClusterEvent (x j) (x (j+1)) s)) ≤ | |
| interpolationConstant (c/2) ε * | |
| (Real.log (1/(G.law p).real (G.twoClusterEvent (x 0) (x d) s))+Nat.clog 2 d+1) := by | |
| let K := (Set.to_countable (Set.univ : Set E)).finiteExhaustion | |
| let δ (n : ℕ) := (G.law p).real {ω | K n ∩ ω ∈ G.twoClusterEvent (x 0) (x d) s} | |
| let δlim := (G.law p).real (G.twoClusterEvent (x 0) (x d) s) | |
| have hδt : Tendsto δ atTop (𝓝 δlim) := G.tendsto_real_twoCluster_restrict p _ _ _ K | |
| have hδpos : ∀ᶠ n in atTop, 0 < δ n := hδt.eventually (eventually_gt_nhds hδ) | |
| have hcc : c/2 < c := by linarith | |
| have hce : ∀ᶠ n in atTop, ∀ j k : Fin (d+1), j.val<k.val → k.val-j.val<d → | |
| c/2 ≤ (G.law p).real {ω | (G.openGraph (K n ∩ ω)).Reachable (x j.val) (x k.val)} := by | |
| apply Filter.eventually_all.mpr | |
| intro j | |
| apply Filter.eventually_all.mpr | |
| intro k | |
| by_cases hjk : j.val<k.val ∧ k.val-j.val<d | |
| · have hl := lt_of_lt_of_le hcc (hconn j k hjk.1 (by omega) hjk.2) | |
| filter_upwards [(G.tendsto_real_reachable_restrict p (x j) (x k) K).eventually (eventually_gt_nhds hl)] | |
| with n hn | |
| exact fun _ _ => hn.le | |
| · exact Filter.Eventually.of_forall (fun _ _ _ => (hjk ⟨by assumption,by assumption⟩).elim) | |
| let C := interpolationConstant (c/2) ε | |
| let r (n : ℕ) := Real.exp (-C*(Real.log (1/δ n)+Nat.clog 2 d+1)) | |
| let b := Real.exp (-C*(Real.log (1/δlim)+Nat.clog 2 d+1)) | |
| have hr : Tendsto r atTop (𝓝 b) := by | |
| have h1 : Tendsto (fun n => 1/δ n) atTop (𝓝 (1/δlim)) := tendsto_const_nhds.div hδt (ne_of_gt hδ) | |
| have h2 := h1.log (one_div_ne_zero (ne_of_gt hδ)) | |
| have hh : Tendsto (fun n => -C*(Real.log (1/δ n)+Nat.clog 2 d+1)) atTop | |
| (𝓝 (-C*(Real.log (1/δlim)+Nat.clog 2 d+1))) := | |
| tendsto_const_nhds.mul ((h2.add_const _).add_const _) | |
| exact (Real.continuous_exp.tendsto _).comp hh | |
| have he : ∀ᶠ n in atTop, ∃ j : Fin d, | |
| r n ≤ (G.law p).real {ω | K n ∩ ω ∈ G.twoClusterEvent (x j.val) (x (j.val+1)) s} := by | |
| filter_upwards [hδpos,hce] with n hnδ hnc | |
| obtain ⟨j,hjd,hjp,hjl⟩ := G.restricted_interpolation_measure (K n) (K.finite n) d x hd p hp | |
| (by linarith : 0<c/2) hε hεp hεq (fun j k hjk hkd hdiff => | |
| hnc ⟨j,by omega⟩ ⟨k,by omega⟩ hjk hdiff) s hnδ | |
| refine ⟨⟨j,hjd⟩,?_⟩ | |
| apply (Real.le_log_iff_exp_le hjp).mp | |
| change -Real.log _ ≤ C * (Real.log (1/δ n)+Nat.clog 2 d+1) at hjl | |
| linarith | |
| obtain ⟨j,hj⟩ := exists_le_of_eventually_exists_le | |
| (fun j : Fin d => G.tendsto_real_twoCluster_restrict p (x j) (x (j.val+1)) s K) hr he | |
| have hjp := lt_of_lt_of_le (Real.exp_pos _) hj | |
| refine ⟨j,j.isLt,hjp,?_⟩ | |
| have hl := (Real.le_log_iff_exp_le hjp).mpr hj | |
| change -C*(Real.log (1/δlim)+Nat.clog 2 d+1) ≤ _ at hl | |
| change -Real.log _ ≤ C * (Real.log (1/δlim)+Nat.clog 2 d+1) | |
| linarith | |
| end BondGraph | |
| end CriticalPercolation | |
| end | |
| end | |
| end | |
| end OAI | |