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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 90 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.HitEvent
import OAI.Probability.CriticalPercolation.Criticality.FiniteEnergy
import OAI.Probability.CriticalPercolation.Gluing.PinnedTest
import OAI.Probability.CriticalPercolation.Gluing.SourceConnectivity
import OAI.Probability.CriticalPercolation.Criticality.RestrictedSeed
import OAI.Probability.CriticalPercolation.ProductMeasure.ProductEmbedding
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
open BernoulliProduct
variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E)
omit [Countable E] in
lemma source_hit_ae_seed (A T : Set V) (q : (E⊕A) → unitInterval)
(hq : ∀ a : A, q (Sum.inr a)=1) :
(G.withSource A).hitEvent none (some '' T) =ᵐ[BernoulliProduct.law q]
((fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω) ⁻¹' G.seedHitEvent A T) := by
have ha (a : A) : ∀ᵐ ω ∂BernoulliProduct.law q, Sum.inr a∈ω := by
rw [ae_iff]
have he : {ω : Set (E⊕A) | ¬Sum.inr a∈ω}={ω | (Sum.inr a∈ω)=False} := by
ext ω; simp
rw [he,law_coordinate,hq a]
simp [bitMeasure]
filter_upwards [ae_all_iff.mpr ha] with ω hω
exact propext (G.source_hit_iff_seedHit A T ω hω)
lemma source_hit_diff_probability (A B T : Set V) (q : (E⊕A) → unitInterval)
(hq : ∀ a : A, q (Sum.inr a)=1) :
BernoulliProduct.law q ((G.withSource A).hitEvent none (some '' B)\
(G.withSource A).hitEvent none (some '' T))=
BernoulliProduct.law (fun b => q (Sum.inl b)) (G.seedHitEvent A B\G.seedHitEvent A T) := by
have heq : BernoulliProduct.law q ((G.withSource A).hitEvent none (some '' B)\
(G.withSource A).hitEvent none (some '' T))=
BernoulliProduct.law q ((fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω) ⁻¹'
(G.seedHitEvent A B\G.seedHitEvent A T)) := by
apply measure_congr
filter_upwards [G.source_hit_ae_seed A B q hq,G.source_hit_ae_seed A T q hq] with ω hB hT
exact propext (and_congr (eq_iff_iff.mp hB) (not_congr (eq_iff_iff.mp hT)))
rw [heq]
change BernoulliProduct.law q ((fun ω : Set (E⊕A) => fun b => Sum.inl b∈ω) ⁻¹' _)=_
rw [← Measure.map_apply (Measurable.of_eval (fun bond => measurable_set_mem (Sum.inl bond)))
((G.measurableSet_seedHitEvent A B).diff (G.measurableSet_seedHitEvent A T))]
change (Measure.map (fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω) (BernoulliProduct.law q)) _=_
rw [law_map_preimage q Sum.inl Sum.inl_injective]
rfl
lemma restricted_seed_diff_probability (A B T : Set V) (q : E → unitInterval) (R : Set E) :
BernoulliProduct.law q (G.restrictedSeed A B R\G.restrictedSeed A T R)=
BernoulliProduct.law (pinnedParameters q Rᶜ ∅) (G.seedHitEvent A B\G.seedHitEvent A T) := by
rw [← forceOn_map_law,Measure.map_apply (measurable_forceOn _ _)
((G.measurableSet_seedHitEvent A B).diff (G.measurableSet_seedHitEvent A T))]
congr 1
have he (ω : Set E) : forceOn Rᶜ ∅ ω=ω∩R := by
ext e
by_cases he : e∈R <;> simp [he]
ext ω
simp only [Set.mem_sdiff,Set.mem_preimage,restrictedSeed,Set.mem_ofPred_eq,he]
lemma restricted_seed_diff_le_source (A B T : Set V) (q : E → unitInterval)
(I F R σ : Set E) (hFR : F⊆R) :
(blockProbability q I (G.restrictedSeed A B F\G.restrictedSeed A T R) σ).toReal ≤
(BernoulliProduct.law (Sum.elim (pinnedParameters (pinnedParameters q I σ) Rᶜ ∅)
(fun _ : A => 1))).real
((G.withSource A).hitEvent none (some '' B)\(G.withSource A).hitEvent none (some '' T)) := by
rw [blockProbability_eq_pinnedLaw q I σ _
((G.restrictedSeed_measurable _ _ _).diff (G.restrictedSeed_measurable _ _ _))]
rw [measureReal_def,G.source_hit_diff_probability A B T _ (fun _ => rfl)]
change (BernoulliProduct.law (pinnedParameters q I σ)).real _ ≤
(BernoulliProduct.law (pinnedParameters (pinnedParameters q I σ) Rᶜ ∅)).real _
rw [measureReal_def,measureReal_def,← G.restricted_seed_diff_probability]
apply ENNReal.toReal_mono (measure_ne_top _ _)
apply measure_mono
exact Set.sdiff_subset_sdiff_left (G.restrictedSeed_mono A B hFR)
end CriticalPercolation.BondGraph
end
end
end
end OAI
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