| import Mathlib |
| 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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|
