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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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.FiniteEnergy
namespace OAI
section
section
noncomputable section
open Set MeasureTheory ProbabilityTheory
open scoped Classical
namespace CriticalPercolation.BernoulliProduct
variable {E I : Type*}
lemma blockSigma_mono {S T : Set E} (h : S ⊆ T) : blockSigma S ≤ blockSigma T := by
apply iSup₂_le
intro e he
exact le_iSup_of_le e (le_iSup_of_le (h he) le_rfl)
lemma independent_disjoint_blocks (q : E → unitInterval) (S T : Set E)
(h : Disjoint S T) : Indep (blockSigma S) (blockSigma T) (law q) :=
indep_iSup_of_disjoint coordinateSigma_le (independent_coordinates q) h
theorem independent_block_family (q : E → unitInterval) (S : I → Set E)
(hS : Pairwise (fun i j => Disjoint (S i) (S j))) :
iIndep (fun i => blockSigma (S i)) (law q) := by
apply (iIndep_iff _ _).mpr
intro t
induction t using Finset.induction_on with
| empty => intro f hf; simp
| @insert i t hi ih =>
intro f hf
let T := ⋃ j ∈ t, S j
have hj : ∀ j ∈ t, blockSigma (S j) ≤ blockSigma T := by
intro j hj
exact blockSigma_mono (fun e he => Set.mem_iUnion₂.mpr ⟨j,hj,he⟩)
have hm : MeasurableSet[blockSigma T] (⋂ j ∈ t, f j) := by
apply MeasurableSet.biInter t.finite_toSet.countable
intro j hjt
exact hj j hjt _ (hf j (Finset.mem_insert_of_mem hjt))
have hdis : Disjoint (S i) T := by
apply Set.disjoint_left.mpr
intro e hei heT
obtain ⟨j,hjt,hje⟩ := Set.mem_iUnion₂.mp heT
exact Set.disjoint_left.mp (hS (by intro h; subst j; exact hi hjt)) hei hje
have hmuli := ((independent_disjoint_blocks q (S i) T hdis).indepSet_of_measurableSet
(hf i (Finset.mem_insert_self i t)) hm).measure_inter_eq_mul
rw [Finset.set_biInter_insert,Finset.prod_insert hi,hmuli]
rw [ih (fun j hj => hf j (Finset.mem_insert_of_mem hj))]
lemma independent_block_events (q : E → unitInterval) (S : I → Set E)
(hS : Pairwise (fun i j => Disjoint (S i) (S j))) (A : I → Set (Set E))
(hA : ∀ i, MeasurableSet[blockSigma (S i)] (A i)) : iIndepSet A (law q) :=
(independent_block_family q S hS).iIndepSets'.iIndepSet_of_mem hA
(fun i => blockSigma_le (S i) _ (hA i))
def closedBlock (S : Set E) : Set (Set E) := {ω | Disjoint S ω}
lemma measurableSet_closed_coordinate (e : E) :
MeasurableSet[coordinateSigma e] {ω : Set E | e ∉ ω} := by
refine ⟨{False},MeasurableSet.singleton _,?_⟩
ext ω
simp only [Set.mem_preimage,Set.mem_singleton_iff,Set.mem_ofPred_eq]
constructor
· intro h hp
exact Eq.mp h hp
· intro h
exact propext (iff_false_intro h)
lemma measurableSet_closedBlock (S : Set E) (hS : S.Countable) :
MeasurableSet[blockSigma S] (closedBlock S) := by
have heq : closedBlock S = ⋂ e ∈ S, {ω : Set E | e ∉ ω} := by
ext ω
simp only [closedBlock,Set.mem_ofPred_eq,Set.mem_iInter,Set.disjoint_left]
rw [heq]
apply MeasurableSet.biInter hS
intro e he
exact (le_iSup_of_le e (le_iSup_of_le he le_rfl) : coordinateSigma e ≤ blockSigma S)
_ (measurableSet_closed_coordinate e)
lemma law_closedBlock (q : E → unitInterval) (S : Set E) (hS : S.Finite) :
law q (closedBlock S) = ∏ e ∈ hS.toFinset, bitMeasure (q e) {False} := by
have heq : closedBlock S = ⋂ e ∈ hS.toFinset, {ω : Set E | (e ∈ ω) = False} := by
ext ω
simp only [closedBlock,Set.mem_ofPred_eq,Set.mem_iInter,Set.disjoint_left,
Set.Finite.mem_toFinset]
constructor
· intro h e he
exact propext (iff_false_intro (h he))
· intro h e he hp
exact Eq.mp (h e he) hp
rw [heq,(independent_coordinates q).meas_biInter]
· exact Finset.prod_congr rfl (fun e _ => law_coordinate q e False)
· intro e _
exact ⟨{False},MeasurableSet.singleton _,rfl⟩
lemma law_closedBlock_const (p : unitInterval) (S : Set E) (hS : S.Finite) :
law (fun _ : E => p) (closedBlock S) =
(bitMeasure p {False}) ^ S.ncard := by
rw [law_closedBlock (fun _ : E => p) S hS,Finset.prod_const]
rw [Set.ncard_eq_toFinset_card S hS]
end CriticalPercolation.BernoulliProduct
end
end
end
end OAI
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