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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace CriticalPercolation.ClusterColumns
open CriticalPercolation.ClusterDeletion
open scoped Classical
variable {V : Type*}
noncomputable def hitValue (A U : Set V) : ℝ := if (U ∩ A).Nonempty then 1 else 0
noncomputable def jointHitValue (A T U : Set V) : ℝ :=
if (U ∩ A).Nonempty ∧ (U ∩ T).Nonempty then 1 else 0
lemma hitValue_nonneg (A U : Set V) : 0 ≤ hitValue A U := by unfold hitValue; positivity
lemma hitValue_le_one (A U : Set V) : hitValue A U ≤ 1 := by unfold hitValue; split_ifs <;> norm_num
lemma hitValue_monotone (A : Set V) : Monotone (hitValue A) := by
intro U W hUW
unfold hitValue
split_ifs with hU hW hW
· exact le_rfl
· exact (hW (hU.mono (Set.inter_subset_inter_left A hUW))).elim
· norm_num
· exact le_rfl
lemma jointHitValue_nonneg (A T U : Set V) : 0 ≤ jointHitValue A T U := by
unfold jointHitValue; positivity
lemma jointHitValue_monotone (A T : Set V) : Monotone (jointHitValue A T) := by
intro U W hUW
unfold jointHitValue
split_ifs with hU hW hW
· exact le_rfl
· exact (hW ⟨hU.1.mono (Set.inter_subset_inter_left A hUW),
hU.2.mono (Set.inter_subset_inter_left T hUW)⟩).elim
· norm_num
· exact le_rfl
lemma jointHitValue_eq_hitValue {A T U : Set V} (hA : (U ∩ A).Nonempty) :
jointHitValue A T U = hitValue T U := by simp [jointHitValue, hitValue, hA]
lemma component_inter_nonempty_iff (H : SimpleGraph V) (s : V) (A : Set V) :
(component H s ∩ A).Nonempty ↔ s ∈ roots H A := by
constructor
· rintro ⟨t, ht, hA⟩
exact ⟨t, hA, ht.symm⟩
· rintro ⟨t, hA, ht⟩
exact ⟨t, ht.symm, hA⟩
end CriticalPercolation.ClusterColumns
end
end
end OAI
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