Thm* E:TaggedEventStruct, tr:Trace(E).
(switch_inv(E) No-dup-send(E))(tr) ![](FONT/eq.png)
( tr':Trace(E). switch_inv(E)(tr') & AD-normal(E)(tr') & (tr adR(E) tr')) | [switch_normal_exists] |
Thm* E:EventStruct, tr:|E| List, ls: ||tr||.
is-send(E)(tr[ls]) ![](FONT/eq.png)
( j: ||tr||. ls < j ![](FONT/eq.png) is-send(E)(tr[j])) ![](FONT/eq.png)
( i,j: ||tr||. i j ![](FONT/eq.png) is-send(E)(tr[j]) ![](FONT/eq.png) (i (switchR(tr)^*) ls) ![](FONT/eq.png) (j (switchR(tr)^*) ls)) | [switch_inv_rel_closure_lemma1] |
Thm* E:TaggedEventStruct, tr:|E| List, ls: ||tr||.
switch_inv(E)(tr) ![](FONT/eq.png)
( i,j: ||tr||. (i (switchR(tr)^*) ls) ![](FONT/eq.png) (j (switchR(tr)^*) ls) ![](FONT/eq.png) tag(E)(tr[i]) = tag(E)(tr[j])) | [switch_inv_rel_closure] |
Thm* E:TaggedEventStruct, tr:|E| List.
switch_inv(E)(tr) ![](FONT/if_big.png) ( i,j: ||tr||. (i switchR(tr) j) ![](FONT/eq.png) tag(E)(tr[i]) = tag(E)(tr[j])) | [switch_inv_rel_same_tag] |
Thm* E:EventStruct, tr:|E| List, ls,i: ||tr||.
is-send(E)(tr[ls]) ![](FONT/eq.png) (i (switchR(tr)^*) ls) ![](FONT/eq.png) is-send(E)(tr[i]) | [switch_inv_rel_closure_send] |
Thm* E:EventStruct, P:((Label![](FONT/dash.png) (|E| List))![](FONT/dash.png) Prop).
( f,g:(Label![](FONT/dash.png) (|E| List)). ( p:Label. g(p) f(p)) ![](FONT/eq.png) P(f) ![](FONT/eq.png) P(g)) ![](FONT/eq.png)
( f,g:(Label![](FONT/dash.png) (|E| List)).
( a:|E|. p:Label. g(p) = filter( b.![](FONT/not.png) (b =msg=(E) a);f(p))) ![](FONT/eq.png) P(f) ![](FONT/eq.png) P(g))
![](FONT/eq.png)
( f,g,h:(Label![](FONT/dash.png) (|E| List)).
( p,q:Label. ( x f(p).( y g(q). (x =msg=(E) y)))) ![](FONT/eq.png)
( p:Label. h(p) = ((f(p)) @ (g(p)))) ![](FONT/eq.png) P(f) ![](FONT/eq.png) P(g) ![](FONT/eq.png) P(h))
![](FONT/eq.png)
switchable0(E)(local_deliver_property(E;P)) | [local_deliver_switchable] |
Thm* E:EventStruct, x:|E| List, j,z: ||x||. Dec(j switchR(x) z) | [decidable__switch_inv_rel] |
Thm* E:EventStruct, x,y:|E| List. (x asyncR(E) y) ![](FONT/eq.png) (y asyncR(E) x) | [R_async_symmetric] |
Thm* E:EventStruct, x,y:|E| List.
(x delayableR(E) y) ![](FONT/eq.png) (y delayableR(E) x) | [R_delayable_symmetric] |
Thm* E:EventStruct, A:Type, evt:(A![](FONT/dash.png) |E|), tg:(A![](FONT/dash.png) Label), m:Label
, tr1,tr2:A List. (tr1 R(tg) tr2) ![](FONT/eq.png) < tr1 > _m = < tr2 > _m A List | [tag_sublist_preserved] |
Thm* E:TaggedEventStruct, P,I,J,K:TraceProperty(E)
, R:(Trace(E)![](FONT/dash.png) Trace(E)![](FONT/dash.png) Prop).
tag_splitable(E;R) ![](FONT/eq.png)
( tr_1,tr_2:Trace(E). (tr_1 R tr_2) ![](FONT/eq.png) (tr_2 R tr_1)) ![](FONT/eq.png)
R preserves P ![](FONT/eq.png)
R preserves K ![](FONT/eq.png)
( tr:Trace(E). (I K)(tr) ![](FONT/eq.png) ( tr':Trace(E). I(tr') & J(tr') & (tr R tr'))) ![](FONT/eq.png)
(((I J) K) fuses P) ![](FONT/eq.png) ((I K) fuses P) | [normal_form_fusion] |
Thm* E:EventStruct, P:TraceProperty(E), L,L1:|E| List.
memorylessR(E) preserves P ![](FONT/eq.png) P(L) ![](FONT/eq.png) P((L -x =msg=(E) y L1)) | [memoryless_remove_msgs] |
Thm* E:EventStruct, tr:|E| List.
No-dup-deliver(E)(tr)
![](FONT/if_big.png)
( x,y:|E|.
is-send(E)(x) ![](FONT/eq.png)
is-send(E)(y) ![](FONT/eq.png) (y =msg=(E) x) ![](FONT/eq.png) loc(E)(x) = loc(E)(y) ![](FONT/eq.png) sublist(|E|;[x; y];tr)) | [P_no_dup_iff] |
Thm* E:EventStruct, tr:|E| List.
Causal(E)(tr) ![](FONT/if_big.png) ( tr':|E| List. tr' tr ![](FONT/eq.png) ( x tr'.( y tr'.is-send(E)(y) & (y =msg=(E) x)))) | [P_causal_iff] |
Def full_switch_inv(E;A;evt;tg;tr_u;tr_l)
== tr_m:A List.
(tr_l R(tg) tr_m) & (map(evt;tr_m) layerR(E) tr_u) & switch_inv( < A,evt,tg > (E))(tr_m) | [full_switch_inv] |
Def delayableR(E)
== swap adjacent[![](FONT/not.png) (x =msg=(E) y)
& ![](FONT/not.png) (is-send(E)(x)) & (is-send(E)(y)) (is-send(E)(x)) & ![](FONT/not.png) (is-send(E)(y))] | [R_delayable] |
Def AD-normal(E)(tr)
== i: (||tr||-1).
( (is-send(E)(tr[i])) ![](FONT/eq.png) ![](FONT/not.png) (is-send(E)(tr[(i+1)])) ![](FONT/eq.png) (tr[i] =msg=(E) tr[(i+1)]))
& (( x,y: ||tr||.
x < y
& (is-send(E)(tr[x]))
& (is-send(E)(tr[y]))
& tr[x] delivered at time i+1
& tr[y] delivered at time i)
![](FONT/eq.png)
loc(E)(tr[i]) = loc(E)(tr[(i+1)])) | [switch_normal] |
Def Macro
x R_del(E) y
== ![](FONT/not.png) (x =msg=(E) y)
& is-deliver(E)(x) & (is-send(E)(y)) (is-send(E)(x)) & is-deliver(E)(y) | [R_del] |
Def memorylessR(E)(L_1,L_2)
== a:|E|. L_2 = filter( b.![](FONT/not.png) (b =msg=(E) a);L_1) |E| List | [R_memoryless] |
Def composableR(E)(L_1,L_2,L)
== ( x L_1.( y L_2.![](FONT/not.png) (x =msg=(E) y))) & L = (L_1 @ L_2) |E| List | [R_composable] |
Def R_ad_normal(tr)(a,b)
== ( (is-send(E)(a)) ![](FONT/eq.png) ![](FONT/not.png) (is-send(E)(b)) ![](FONT/eq.png) (a =msg=(E) b))
& (![](FONT/not.png) (is-send(E)(a)) ![](FONT/eq.png)
![](FONT/not.png) (is-send(E)(b)) ![](FONT/eq.png)
( x,y: ||tr||.
x < y
& (is-send(E)(tr[x]))
& (is-send(E)(tr[y]))
& (tr[x] =msg=(E) b)
& (tr[y] =msg=(E) a))
![](FONT/eq.png)
loc(E)(a) = loc(E)(b)) | [R_ad_normal] |
Def single-tag-decomposable(E)(L)
== L = nil |E| List ![](FONT/eq.png)
( L_1,L_2:Trace(E).
L = (L_1 @ L_2) |E| List
& L_2 = nil |E| List
& ( x L_1.( y L_2.![](FONT/not.png) (x =msg=(E) y)))
& ( m:Label. ( x L_2.tag(E)(x) = m))) | [single_tag_decomposable] |
Def C(Q)(i) == k: ||L||. Q(k) & (L[k] =msg=(E) L[i]) | [message_closure] |
Def Causal(E)(tr)
== i: ||tr||. j: ||tr||. j i & (is-send(E)(tr[j])) & (tr[j] =msg=(E) tr[i]) | [P_causal] |
Def No-dup-deliver(E)(tr)
== i,j: ||tr||.
![](FONT/not.png) (is-send(E)(tr[i])) ![](FONT/eq.png)
![](FONT/not.png) (is-send(E)(tr[j])) ![](FONT/eq.png) (tr[j] =msg=(E) tr[i]) ![](FONT/eq.png) loc(E)(tr[i]) = loc(E)(tr[j]) ![](FONT/eq.png) i = j | [P_no_dup] |
Def Tag-by-msg(E)(tr)
== i,j: ||tr||. (tr[i] =msg=(E) tr[j]) ![](FONT/eq.png) tag(E)(tr[i]) = tag(E)(tr[j]) | [P_tag_by_msg] |
Def x delivered at time k == (x =msg=(E) tr[k]) & ![](FONT/not.png) (is-send(E)(tr[k])) | [delivered_at] |
Def switch_inv(E; tr)
== i,j,k: ||tr||.
i < j ![](FONT/eq.png)
(is-send(E)(tr[i])) ![](FONT/eq.png)
(is-send(E)(tr[j])) ![](FONT/eq.png)
tag(E)(tr[i]) = tag(E)(tr[j]) ![](FONT/eq.png)
(tr[j] =msg=(E) tr[k]) ![](FONT/eq.png)
![](FONT/not.png) (is-send(E)(tr[k])) ![](FONT/eq.png)
( k': ||tr||.
k' < k & loc(E)(tr[k']) = loc(E)(tr[k]) & (tr[i] =msg=(E) tr[k']) & ![](FONT/not.png) (is-send(E)(tr[k']))) | [switch_inv2001_03_15_DASH_PM_DASH_12_53_21] |