File size: 5,459 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.ProductMeasure.TailProduct
import OAI.Probability.CriticalPercolation.ProductMeasure.LevelContactMeasurability
import OAI.Probability.CriticalPercolation.Gluing.HonestChoice
import OAI.Probability.CriticalPercolation.Gluing.TrialSelection
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory
open scoped Classical BigOperators
namespace CriticalPercolation
lemma measurableSet_finset_subset {Ω V : Type*} {m : MeasurableSpace Ω}
(S : Ω → Set V) (hS : @Measurable Ω (Set V) m inferInstance S) (U : Finset V) :
MeasurableSet[m] {ω | (U : Set V)⊆S ω} := by
have he : {ω | (U : Set V)⊆S ω}=⋂ v∈U, {ω | v∈S ω} := by
ext ω
simp [Set.subset_def]
rw [he]
exact MeasurableSet.biInter U.countable_toSet (fun v _ =>
((measurable_pi_apply v).comp hS).setOf)
namespace BondGraph
open BernoulliProduct
variable {V E : Type*} (G : BondGraph V E)
lemma levelContacts_reachable (π : V → ℤ) (j : ℤ) (o : V) (ω : Set E)
{v : V} (hv : v∈G.levelContacts π j o ω) : (G.openGraph ω).Reachable o v :=
G.cluster_mono Set.inter_subset_left o hv.1
lemma levelContacts_nonempty_of_hit (π : V → ℤ) (j : ℤ) (o : V) (ho : j ≤ π o)
(hcross : ∀ e v w, G.ends e=s(v,w) → j ≤ π v → π w<j → π v=j)
(ω : Set E) (hhit : ω∈G.hitEvent o {v | π v<j}) :
(G.levelContacts π j o ω).Nonempty := by
by_contra h
have hd : G.contactDrop π j o ω=∅ := by
apply Set.eq_empty_iff_forall_notMem.mpr
rintro e ⟨v,hv,w,hw,hend⟩
exact h ⟨v,hv⟩
have hc : Disjoint (G.contactDrop π j o ω) ω := by rw [hd]; exact Set.empty_disjoint _
have he := G.cluster_eq_exterior_of_drop_closed π j o ho hcross ω hc
obtain ⟨v,hv,hlt⟩ := hhit
have hle : j ≤ π v := G.exteriorCluster_subset_level π j o ho ω (he ▸ hv)
exact (not_le.mpr hlt) hle
lemma levelContacts_supply (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ) (ω : Set E)
(hne : (G.levelContacts π j o ω).Nonempty) (hfew : ω∉G.fewContacts π j o k) :
∃ S : Finset V, (S : Set V)⊆G.levelContacts π j o ω ∧ k ≤ S.card := by
by_cases hfin : (G.levelContacts π j o ω).Finite
· refine ⟨hfin.toFinset,by simp,?_⟩
by_contra hcard
exact hfew ⟨hfin.toFinset,hfin.coe_toFinset.symm,
by simpa using hne,lt_of_not_ge hcard⟩
· obtain ⟨S,hS,hSfin,hcard⟩ := Set.Infinite.exists_subset_ncard_eq hfin k
exact ⟨hSfin.toFinset,by simpa using hS,by simpa [Set.ncard_eq_toFinset_card S hSfin] using hcard.ge⟩
variable [Countable V] [Countable E]
def manyContacts (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ) : Set (Set E) :=
{ω | (G.levelContacts π j o ω).Nonempty}\G.fewContacts π j o k
lemma measurableSet_manyContacts (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ) :
MeasurableSet[blockSigma (G.inducedEdges {v | j ≤ π v})] (G.manyContacts π j o k) := by
apply MeasurableSet.diff _ (G.measurableSet_fewContacts π j o k)
have he : {ω | (G.levelContacts π j o ω).Nonempty}=⋃ v : V, {ω | v∈G.levelContacts π j o ω} := by
ext ω
simp [Set.Nonempty]
rw [he]
exact MeasurableSet.iUnion (fun v =>
@Measurable.setOf (Set E) (blockSigma (G.inducedEdges {v | j ≤ π v})) _
((measurable_pi_apply v).comp (G.measurable_levelContacts π j o)))
theorem honest_contact_selection (π : V → ℤ) (j : ℤ) (o : V) (k : ℕ)
(C : V → V → Prop) (hrefl : ∀ v, C v v) (hsymm : ∀ {v w}, C v w → C w v)
(M N : ℕ) (hM : 0 < M) (hk : M*N ≤ k)
(hfinite : ∀ v, {w | C v w}.Finite) (hbound : ∀ v, {w | C v w}.ncard ≤ M)
(s : V → ℝ) (s₀ K : ℝ) (hK : 0 ≤ K)
(hs : ∀ v, π v=j → s₀ ≤ s v ∧ s v ≤ 1) (hN : K ≤ N*s₀) :
∃ d : Set E → Finset V,
(∀ U, MeasurableSet[blockSigma (G.inducedEdges {v | j ≤ π v})] {ω | d ω=U}) ∧
∀ ω∈G.manyContacts π j o k,
(d ω : Set V)⊆G.levelContacts π j o ω ∧
(d ω : Set V).Pairwise (fun v w => ¬C v w) ∧
K ≤ ∑ v∈d ω, s v ∧ (∑ v∈d ω, s v) ≤ K+1 := by
let P := fun (ω : Set E) (U : Finset V) =>
(U : Set V)⊆G.levelContacts π j o ω ∧
(U : Set V).Pairwise (fun v w => ¬C v w) ∧
K ≤ ∑ v∈U, s v ∧ (∑ v∈U, s v) ≤ K+1
have hm (U : Finset V) : MeasurableSet[blockSigma (G.inducedEdges {v | j ≤ π v})] {ω | P ω U} :=
(measurableSet_finset_subset _ (G.measurable_levelContacts π j o) U).inter (MeasurableSet.const _)
obtain ⟨d,hd,hgood⟩ := @exists_honest_selector (Set E) (Finset V)
(blockSigma (G.inducedEdges {v | j ≤ π v})) inferInstance inferInstance P hm
refine ⟨d,hd,?_⟩
intro ω hω
apply hgood
obtain ⟨S,hS,hcard⟩ := G.levelContacts_supply π j o k ω hω.1 hω.2
have hb : ∀ v∈S, (S.filter (C v)).card ≤ M := by
intro v _hv
have hh : (S.filter (C v) : Set V)⊆{w | C v w} := by
intro w hw
exact (Finset.mem_filter.mp hw).2
have hz := (Set.ncard_le_ncard hh (hfinite v)).trans (hbound v)
simpa only [Set.ncard_coe_finset] using hz
obtain ⟨U,hU,hsep,hlo,hhi⟩ := separated_weighted_trials C hrefl hsymm M N hM S hb
(hk.trans hcard) s s₀ K hK (fun v hv => hs v (hS hv).2) hN
exact ⟨U,(fun _ hv => hS (hU hv)),hsep,hlo,hhi⟩
end BondGraph
end CriticalPercolation
end
end
end
end OAI
|