Thm* E:TaggedEventStruct, tr:|E| List, ls: ||tr||.
switch_inv(E)(tr) 
( i,j: ||tr||. (i (switchR(tr)^*) ls)  (j (switchR(tr)^*) ls)  tag(E)(tr[i]) = tag(E)(tr[j])) | [switch_inv_rel_closure] |
Thm* E:TaggedEventStruct, tr:|E| List.
switch_inv(E)(tr)  ( i,j: ||tr||. (i switchR(tr) j)  tag(E)(tr[i]) = tag(E)(tr[j])) | [switch_inv_rel_same_tag] |
Def switch_inv(E)(tr)
== i,j,k: ||tr||.
i < j 
(is-send(E)(tr[i])) 
(is-send(E)(tr[j])) 
tag(E)(tr[i]) = tag(E)(tr[j]) 
tr[j] delivered at time k 
( k': ||tr||. k' < k & tr[i] delivered at time k' & loc(E)(tr[k']) = loc(E)(tr[k])) | [switch_inv] |
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 single-tag-decomposable(E)(L)
== L = nil |E| List 
( L_1,L_2:Trace(E).
L = (L_1 @ L_2) |E| List
& L_2 = nil |E| List
& ( x L_1.( y L_2. (x =msg=(E) y)))
& ( m:Label. ( x L_2.tag(E)(x) = m))) | [single_tag_decomposable] |
Def Tag-by-msg(E)(tr)
== i,j: ||tr||. (tr[i] =msg=(E) tr[j])  tag(E)(tr[i]) = tag(E)(tr[j]) | [P_tag_by_msg] |
Def switch_inv(E; tr)
== i,j,k: ||tr||.
i < j 
(is-send(E)(tr[i])) 
(is-send(E)(tr[j])) 
tag(E)(tr[i]) = tag(E)(tr[j]) 
(tr[j] =msg=(E) tr[k]) 
 (is-send(E)(tr[k])) 
( k': ||tr||.
k' < k & loc(E)(tr[k']) = loc(E)(tr[k]) & (tr[i] =msg=(E) tr[k']) &  (is-send(E)(tr[k']))) | [switch_inv2001_03_15_DASH_PM_DASH_12_53_21] |