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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.Score
import OAI.Probability.CriticalPercolation.Modular.ModularClusterCount
import OAI.Probability.CriticalPercolation.Modular.HeavyClusterNumber
import OAI.Probability.CriticalPercolation.ProductMeasure.ProductConditioning
import OAI.Probability.CriticalPercolation.Clusters.FiniteForcing
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma modularMass_union_ne_top (o : V) (S T : Set V) :
G.modularMass o (S∪T)≠∞ ↔ G.modularMass o S≠∞ ∧ G.modularMass o T≠∞ := by
constructor
· intro h
exact ⟨ne_top_of_le_ne_top h (G.modularMass_mono o Set.subset_union_left),
ne_top_of_le_ne_top h (G.modularMass_mono o Set.subset_union_right)⟩
· rintro ⟨hS,hT⟩
exact ne_top_of_le_ne_top (ENNReal.add_ne_top.mpr ⟨hS,hT⟩) (G.modularMass_union_le o S T)
lemma finiteMassIndicator_insert (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(ω : Set E) (e : E) (a b : V) (he : G.ends e=s(a,b)) :
G.finiteMassIndicator (insert e ω) a=G.finiteMassIndicator ω a*G.finiteMassIndicator ω b := by
have hh : G.modularMass a (G.cluster (insert e ω) a)≠∞ ↔
G.modularMass a (G.cluster ω a)≠∞ ∧ G.modularMass b (G.cluster ω b)≠∞ := by
rw [G.cluster_insert_endpoint ω e a b he,G.modularMass_union_ne_top]
exact and_congr_right (fun _ => not_congr (G.modularMass_eq_top_change_base hc hl a b (G.cluster ω b)))
by_cases ha : G.modularMass a (G.cluster ω a)=∞ <;>
by_cases hb : G.modularMass b (G.cluster ω b)=∞
· have hi : G.modularMass a (G.cluster (insert e ω) a)=∞ := by tauto
simp [finiteMassIndicator,ha,hb,hi]
· have hi : G.modularMass a (G.cluster (insert e ω) a)=∞ := by tauto
simp [finiteMassIndicator,ha,hb,hi]
· have hi : G.modularMass a (G.cluster (insert e ω) a)=∞ := by tauto
simp [finiteMassIndicator,ha,hb,hi]
· have hi := hh.mpr ⟨ha,hb⟩
simp [finiteMassIndicator,ha,hb,hi]
lemma incidentClusterCount_insert_of_ends (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(ω : Set E) (e : E) (a b : V) (he : G.ends e=s(a,b)) :
G.incidentClusterCount (insert e ω) e=G.finiteMassIndicator ω a*G.finiteMassIndicator ω b := by
rw [G.incidentClusterCount_of_ends hc hl _ e a b he,
ite_eq_left (LazyDFS.prop_openEdge_reachable G (insert e ω) (Set.mem_insert e ω) he),
G.finiteMassIndicator_insert hc hl ω e a b he]
lemma incidentClusterCount_flip_of_ends (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(ω : Set E) (e : E) (a b : V) (he : G.ends e=s(a,b)) :
(G.incidentClusterCount ω e).toReal-(G.incidentClusterCount (insert e ω) e).toReal =
if ¬(G.openGraph ω).Reachable a b ∧
¬(G.modularMass a (G.cluster ω a)=∞ ∧ G.modularMass b (G.cluster ω b)=∞) then 1 else 0 := by
rw [G.incidentClusterCount_of_ends hc hl ω e a b he,G.incidentClusterCount_insert_of_ends hc hl ω e a b he]
by_cases hr : (G.openGraph ω).Reachable a b
· have hf := G.finiteMassIndicator_eq_of_reachable hc hl ω hr
rw [ite_eq_left hr,←hf]
unfold finiteMassIndicator
split_ifs <;> simp_all
· simp only [ite_eq_right hr]
unfold finiteMassIndicator
by_cases ha : G.modularMass a (G.cluster ω a)=∞ <;>
by_cases hb : G.modularMass b (G.cluster ω b)=∞ <;> norm_num [ha,hb,hr]
lemma forceOn_univ_singleton (e : E) (ω : Set E) :
BernoulliProduct.forceOn {e} Set.univ ω=insert e ω := by
ext f
by_cases hf : f=e <;> simp [hf]
lemma forceOn_empty_singleton (e : E) (ω : Set E) :
BernoulliProduct.forceOn {e} ∅ ω=ω\{e} := BernoulliProduct.forceOn_empty_eq_sdiff {e} ω
end CriticalPercolation.BondGraph
end
end
end OAI
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