File size: 2,906 Bytes
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import OAI.Probability.CriticalPercolation.Gluing.PredictionFailure
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*}
lemma blockProbability_eq_pinnedLaw (q : E → unitInterval) (I σ : Set E)
(A : Set (Set E)) (hA : MeasurableSet A) :
blockProbability q I A σ=law (pinnedParameters q I σ) A := by
rw [← forceOn_map_law,Measure.map_apply (measurable_forceOn I σ) hA]
rfl
def predictionFailure (q : E → unitInterval) (F : Set E)
(A C D : Set (Set E)) (δ : ℝ) : Set (Set E) :=
Aᶜ ∪ (A∩{σ | (blockProbability q F C σ).toReal<1-δ}) ∪
(A∩{σ | (blockProbability q F D σ).toReal<1-δ})
lemma measurableSet_predictionFailure (q : E → unitInterval) (F : Set E)
(A C D : Set (Set E)) (hA : MeasurableSet[blockSigma F] A)
(hC : MeasurableSet C) (hD : MeasurableSet D) (δ : ℝ) :
MeasurableSet[blockSigma F] (predictionFailure q F A C D δ) := by
exact (hA.compl.union (hA.inter (measurableSet_lt
(measurable_blockProbability q F C hC).ennreal_toReal measurable_const))).union
(hA.inter (measurableSet_lt
(measurable_blockProbability q F D hD).ennreal_toReal measurable_const))
theorem pinned_prediction_failure (q : E → unitInterval) (I F σ : Set E)
(hIF : I⊆F) (A C D : Set (Set E))
(hA : MeasurableSet[blockSigma F] A) (hC : MeasurableSet C) (hD : MeasurableSet D)
(δ a : ℝ) (hδ : 0<δ)
(hin : 1-δ ≤ (blockProbability q I A σ).toReal)
(hCfail : (blockProbability q I (A\C) σ).toReal ≤ a)
(hDfail : (blockProbability q I (A\D) σ).toReal ≤ a) :
(blockProbability q I (predictionFailure q F A C D δ) σ).toReal ≤ δ+2*a/δ := by
have hma := (blockSigma_le F) _ hA
rw [blockProbability_eq_pinnedLaw q I σ A hma] at hin
rw [blockProbability_eq_pinnedLaw q I σ (A\C) (hma.diff hC)] at hCfail
rw [blockProbability_eq_pinnedLaw q I σ (A\D) (hma.diff hD)] at hDfail
have hh := two_prediction_failure (pinnedParameters q I σ) F A C D hA hC hD δ a hδ
hin hCfail hDfail
simp_rw [blockProbability_pin_smaller q I F σ hIF C hC,
blockProbability_pin_smaller q I F σ hIF D hD] at hh
rw [blockProbability_eq_pinnedLaw q I σ _
((blockSigma_le F) _ (measurableSet_predictionFailure q F A C D hA hC hD δ))]
exact hh
lemma predictionFailure_pin_smaller (q : E → unitInterval) (I F σ : Set E)
(hIF : I⊆F) (A C D : Set (Set E)) (hC : MeasurableSet C)
(hD : MeasurableSet D) (δ : ℝ) :
predictionFailure (pinnedParameters q I σ) F A C D δ=predictionFailure q F A C D δ := by
unfold predictionFailure
simp_rw [blockProbability_pin_smaller q I F σ hIF C hC,
blockProbability_pin_smaller q I F σ hIF D hD]
end CriticalPercolation.BernoulliProduct
end
end
end
end OAI
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