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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.BlockIndependence
import OAI.Probability.CriticalPercolation.Gluing.SelectedFresh
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical BigOperators ENNReal
namespace CriticalPercolation.BernoulliProduct
variable {E I : Type*}
theorem selected_fresh_lower [Countable I] (q : E → unitInterval) (X Y : Set E)
(hXY : Disjoint X Y) (d : Set E → I) (A : I → Set (Set E))
(hd : ∀ i, MeasurableSet[blockSigma X] {ω | d ω=i})
(hA : ∀ i, MeasurableSet[blockSigma Y] (A i))
(C : Set (Set E)) (hC : MeasurableSet[blockSigma X] C)
(r : ℝ) (hr : 0 ≤ r) (hb : ∀ σ∈C, r ≤ (law q).real (A (d σ))) :
r*(law q).real C ≤ (law q).real (C∩selectedEvent d A) := by
have hmA := measurableSet_selectedEvent d A (fun i => blockSigma_le X _ (hd i))
(fun i => blockSigma_le Y _ (hA i))
have hk (σ : Set E) (hσ : σ∈C) :
ENNReal.ofReal r ≤ blockProbability q X (selectedEvent d A) σ := by
rw [blockProbability_selected q X Y hXY d A hd hA σ]
apply (ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (measure_ne_top _ _)).mp
rw [ENNReal.toReal_ofReal hr]
exact hb σ hσ
have hh : ENNReal.ofReal r*law q C ≤ law q (selectedEvent d A∩C) := by
rw [← setLIntegral_blockProbability q X _ C hmA hC]
calc
_ = ∫⁻ _ in C, ENNReal.ofReal r ∂law q := by
rw [lintegral_const,Measure.restrict_apply_univ]
_ ≤ _ := setLIntegral_mono' (blockSigma_le X _ hC) hk
have ht := ENNReal.toReal_mono (measure_ne_top _ _) hh
simpa only [Set.inter_comm (selectedEvent d A) C,ENNReal.toReal_mul,
ENNReal.toReal_ofReal hr,Measure.real] using ht
lemma law_closedBlock_lower (q : E → unitInterval) (S : Set E) (hS : S.Finite)
(p : unitInterval) (N : ℕ) (hN : S.ncard ≤ N)
(hq : ∀ e∈S, (q e : ℝ) ≤ p) :
(1-(p : ℝ))^N ≤ (law q).real (closedBlock S) := by
have hp0 : 0 ≤ 1-(p : ℝ) := sub_nonneg.mpr p.2.2
have hp1 : 1-(p : ℝ) ≤ 1 := by linarith [p.2.1]
rw [measureReal_def,law_closedBlock q S hS,ENNReal.toReal_prod]
calc
_ ≤ (1-(p : ℝ))^S.ncard := pow_le_pow_of_le_one hp0 hp1 hN
_ = ∏ e∈hS.toFinset, (1-(p : ℝ)) := by simp [Set.ncard_eq_toFinset_card S hS]
_ ≤ _ := by
apply Finset.prod_le_prod₀
· intro e he; exact hp0
· intro e he
change 1-(p : ℝ) ≤ (bitMeasure (q e)).real {False}
rw [bitMeasure_real_singleton]
simpa only [bitWeight,ite_false] using sub_le_sub_left (hq e (hS.mem_toFinset.mp he)) 1
lemma exists_level_small (μ : Measure (Set E)) [IsProbabilityMeasure μ]
(J : Finset I) (hJ : J.Nonempty) (C M : I → Set (Set E))
(hM : ∀ i∈J, MeasurableSet (M i))
(hdis : (J : Set I).PairwiseDisjoint M) (c : ℝ) (hc : 0 < c)
(hb : ∀ i∈J, c*μ.real (C i) ≤ μ.real (M i)) :
∃ i∈J, μ.real (C i) ≤ 1/((J.card : ℝ)*c) := by
have hsum : (∑ i∈J, μ.real (M i)) ≤ 1 := by
rw [← measureReal_biUnion_finset hdis hM]
exact measureReal_le_one
have hcard : 0 < (J.card : ℝ) := by exact_mod_cast Finset.card_pos.mpr hJ
by_contra h
push Not at h
have hs : (∑ i∈J, 1/(J.card : ℝ)) < ∑ i∈J, μ.real (M i) := by
apply Finset.sum_lt_sum_of_nonempty hJ
intro i hi
have hh := mul_lt_mul_of_pos_left (h i hi) hc
have he : c*(1/((J.card : ℝ)*c))=1/(J.card : ℝ) := by field_simp
rw [he] at hh
exact hh.trans_le (hb i hi)
simp only [Finset.sum_const, nsmul_eq_mul] at hs
have he : (J.card : ℝ)*(1/(J.card : ℝ))=1 := by field_simp
rw [he] at hs
exact (not_lt_of_ge hsum) hs
end CriticalPercolation.BernoulliProduct
end
end
end
end OAI
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