Thm* E:EventStruct, tr:|E| List, ls: ||tr||.
is-send(E)(tr[ls]) 
( j: ||tr||. ls < j  is-send(E)(tr[j])) 
( i,j: ||tr||. i j  is-send(E)(tr[j])  (i (switchR(tr)^*) ls)  (j (switchR(tr)^*) ls)) | [switch_inv_rel_closure_lemma1] |
Def switch-decomposable(E)(L)
== L = nil |E| List
( Q:( ||L|| Prop).
( i: ||L||. Dec(Q(i)))
& ( i: ||L||. Q(i))
& ( i: ||L||. Q(i)  (is-send(E)(L[i])))
& ( i,j: ||L||. Q(i)  Q(j)  tag(E)(L[i]) = tag(E)(L[j]))
& ( i,j: ||L||. Q(i)  i j  C(Q)(j))) | [switch_decomposable] |
Def x somewhere delivered before y
== k: ||tr||.
x delivered at time k
& ( k': ||tr||. y delivered at time k'  loc(E)(tr[k']) = loc(E)(tr[k])  k k') | [delivered_before_somewhere] |
Def Causal(E)(tr)
== i: ||tr||. j: ||tr||. j i & (is-send(E)(tr[j])) & (tr[j] =msg=(E) tr[i]) | [P_causal] |