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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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
namespace CriticalPercolation.ClusterDeletion
variable {V : Type*}
lemma component_self (H : SimpleGraph V) (s : V) : s ∈ component H s :=
SimpleGraph.Reachable.refl s
lemma mem_component_symm {H : SimpleGraph V} {s t : V} :
t ∈ component H s ↔ s ∈ component H t :=
⟨SimpleGraph.Reachable.symm, SimpleGraph.Reachable.symm⟩
lemma roots_singleton (H : SimpleGraph V) (s : V) : roots H {s} = component H s := by
ext v
simp [roots, component]
lemma component_eq_of_mem {H : SimpleGraph V} {s t : V}
(h : t ∈ component H s) : component H t = component H s := by
ext v
exact ⟨fun hv => h.trans hv, fun hv => h.symm.trans hv⟩
lemma not_mem_roots_iff_disjoint (H : SimpleGraph V) (S : Set V) (s : V) :
s ∉ roots H S ↔ Disjoint (component H s) S := by
rw [Set.disjoint_left]
constructor
· intro hs t ht hS
exact hs ⟨t, hS, ht.symm⟩
· intro hd ⟨t, ht, hts⟩
exact hd hts.symm ht
lemma component_delete_component (H : SimpleGraph V) (s t : V)
(h : t ∉ component H s) :
component (delete H (component H s)) t = component H t := by
simpa only [roots_singleton] using
(delete_roots_component H {s} t (by simpa only [roots_singleton] using h))
lemma roots_delete_component (H : SimpleGraph V) (S : Set V) (s : V)
(h : s ∉ roots H S) : roots (delete H (component H s)) S = roots H S := by
ext v
constructor
· intro hv
exact roots_mono (delete_le H _) S hv
· rintro ⟨t, ht, htv⟩
refine ⟨t, ht, ?_⟩
have hts : t ∉ component H s := fun hg => h ⟨t, ht, hg.symm⟩
change v ∈ component (delete H (component H s)) t
rw [component_delete_component H s t hts]
exact htv
lemma delete_empty (H : SimpleGraph V) : delete H ∅ = H := by
ext v w
simp only [delete, Set.mem_empty_iff_false, not_false_eq_true, and_true]
lemma component_delete_self (H : SimpleGraph V) (U : Set V) (s : V)
(hs : s ∈ U) : component (delete H U) s = {s} := by
ext v
constructor
· intro h
have hadj : ∀ t, ¬ (delete H U).Adj s t := fun t ht => ht.2.1 hs
have he : v = s := by
rcases h with ⟨p⟩
cases p with
| nil => rfl
| cons hst _ => exact (hadj _ hst).elim
exact he
· rintro rfl
exact component_self _ _
end CriticalPercolation.ClusterDeletion
end
end
end OAI
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