openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Exploration /FrontierDivergence.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.Criticality.CylinderApproximation | |
| import OAI.Probability.CriticalPercolation.Exploration.SliceExploration | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set MeasureTheory ProbabilityTheory Filter | |
| open scoped Classical ENNReal symmDiff Topology | |
| namespace CriticalPercolation.BondGraph | |
| open BernoulliProduct | |
| variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E) | |
| lemma smallFreshBoundary_approx_bound (hl : G.LocallyFinite) (p : unitInterval) | |
| (X : Set E) (v : V) (M : ℕ) (H : Set (Set E)) | |
| (hH : MeasurableSet[blockSigma X] H) : | |
| (1-(p:ℝ))^M * (G.law p).real (G.infiniteEvent v ∩ G.smallFreshBoundary X v M) ≤ | |
| (1 + (1-(p:ℝ))^M) * (G.law p).real (H ∆ G.infiniteEvent v) := by | |
| have hcost := G.smallFreshBoundary_cost hl p X v M H hH | |
| have hcover : G.infiniteEvent v ∩ G.smallFreshBoundary X v M ⊆ | |
| (H ∩ G.smallFreshBoundary X v M) ∪ (H ∆ G.infiniteEvent v) := by | |
| intro ω hω | |
| by_cases h : ω∈H | |
| · exact Or.inl ⟨h,hω.2⟩ | |
| · exact Or.inr (Or.inr ⟨hω.1,h⟩) | |
| have hm := (measureReal_mono hcover (μ := G.law p)).trans | |
| (measureReal_union_le _ _) | |
| have he : (G.law p).real (H \ G.infiniteEvent v) ≤ | |
| (G.law p).real (H ∆ G.infiniteEvent v) := | |
| measureReal_mono (by intro ω hω; exact Or.inl hω) | |
| have hr : 0 ≤ (1-(p:ℝ))^M := pow_nonneg (sub_nonneg.mpr p.2.2) _ | |
| nlinarith only [hcost,hm,he,hr] | |
| theorem smallFreshBoundary_probability_tendsto_zero | |
| (hl : G.LocallyFinite) (p : unitInterval) (hp : (p:ℝ)<1) | |
| (X : ℕ → Set E) (hX : ∀ S : Finset E, ∀ᶠ n in atTop, (S : Set E) ⊆ X n) | |
| (v : V) (M : ℕ) : | |
| Tendsto (fun n => G.law p (G.infiniteEvent v ∩ G.smallFreshBoundary (X n) v M)) | |
| atTop (𝓝 0) := by | |
| let r : ℝ := (1-(p:ℝ))^M | |
| have hr : 0 < r := pow_pos (sub_pos.mpr hp) _ | |
| have ht : Tendsto (fun n => (G.law p).real | |
| (G.infiniteEvent v ∩ G.smallFreshBoundary (X n) v M)) atTop (𝓝 0) := by | |
| apply tendsto_order.mpr | |
| constructor | |
| · intro a ha | |
| exact Eventually.of_forall (fun _ => lt_of_lt_of_le ha measureReal_nonneg) | |
| · intro b hb | |
| have hδ : 0 < b*r/(1+r) := div_pos (mul_pos hb hr) (by linarith) | |
| obtain ⟨S,H,hH,happrox⟩ := exists_finiteCoordinate_approx (fun _ : E => p) | |
| (G.measurableSet_infiniteEvent v) (ENNReal.ofReal_pos.mpr hδ) | |
| have hreal : (G.law p).real (H ∆ G.infiniteEvent v) < b*r/(1+r) := by | |
| have hh := (ENNReal.toReal_lt_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mpr happrox | |
| simpa only [law_const,ENNReal.toReal_ofReal hδ.le,Measure.real,law] using hh | |
| filter_upwards [hX S] with n hn | |
| have hh := G.smallFreshBoundary_approx_bound hl p (X n) v M H | |
| (blockSigma_mono hn _ hH) | |
| change r * (G.law p).real (G.infiniteEvent v ∩ G.smallFreshBoundary (X n) v M) ≤ | |
| (1+r) * (G.law p).real (H ∆ G.infiniteEvent v) at hh | |
| have hmul := (mul_lt_mul_of_pos_left hreal (show 0<1+r by linarith)) | |
| rw [mul_div_cancel₀ _ (ne_of_gt (show 0<1+r by linarith))] at hmul | |
| nlinarith only [hh,hmul,hr] | |
| have hh := ENNReal.tendsto_ofReal ht | |
| simpa only [ENNReal.ofReal_zero,Measure.real,ENNReal.ofReal_toReal (measure_ne_top _ _)] using hh | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end OAI | |