| Who Cites full switch inv? |
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full_switch_inv | 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) |
| | Thm* E:EventStruct, A:Type, f:(A|E|), t:(ALabel), tr_u:|E| List, tr_l:A List.
full_switch_inv(E;A;f;t;tr_u;tr_l) Prop |
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induced_tagged_event_str |
Def < A,evt,tg > (E)
== < A,MS(E),msg(E) o evt,loc(E) o evt,is-send(E) o evt,tg, > |
| | Thm* E:EventStruct, A:Type, f:(A|E|), t:(ALabel). < A,f,t > (E) TaggedEventStruct |
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switch_inv |
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])) |
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Thm* E:TaggedEventStruct. switch_inv(E) (|E| List)Prop |
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map |
Def map(f;as) == Case of as; nil nil ; a.as' [(f(a)) / map(f;as')] (recursive) |
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Thm* A,B:Type, f:(AB), l:A List. map(f;l) B List |
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Thm* A,B:Type, f:(AB), l:A List. map(f;l) B List |
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layer_rel |
Def layerR(E) == ((asyncR(E) delayableR(E)) send-enabledR(E))^* |
| | Thm* E:EventStruct. layerR(E) (|E| List)(|E| List)Prop |
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tag_rel |
Def R(tg) == swap adjacent[tg(x) = tg(y) Label]^* |
| | Thm* A:Type, tg:(ALabel). R(tg) (A List)(A List)Prop |
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delivered_at |
Def x delivered at time k == (x =msg=(E) tr[k]) & (is-send(E)(tr[k])) |
| | Thm* E:EventStruct, tr:|E| List, x:|E|, k:||tr||. x delivered at time k Prop |
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R_send_enabled |
Def send-enabledR(E)(L_1,L_2) == x:|E|. (is-send(E)(x)) & L_2 = (L_1 @ [x]) |
| | Thm* E:EventStruct. send-enabledR(E) (|E| List)(|E| List)Prop |
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R_delayable |
Def delayableR(E)
== swap adjacent[(x =msg=(E) y)
& (is-send(E)(x)) & (is-send(E)(y)) (is-send(E)(x)) & (is-send(E)(y))] |
| | Thm* E:EventStruct. delayableR(E) (|E| List)(|E| List)Prop |
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R_async |
Def asyncR(E)
== swap adjacent[loc(E)(x) = loc(E)(y)
& (is-send(E)(x)) & (is-send(E)(y)) (is-send(E)(x)) & (is-send(E)(y))] |
| | Thm* E:EventStruct. asyncR(E) (|E| List)(|E| List)Prop |
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event_is_snd |
Def is-send(E) == 1of(2of(2of(2of(2of(E))))) |
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Thm* E:EventStruct. is-send(E) |E| |
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compose |
Def (f o g)(x) == f(g(x)) |
| | Thm* A,B,C:Type, f:(BC), g:(AB). f o g AC |
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event_loc |
Def loc(E) == 1of(2of(2of(2of(E)))) |
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Thm* E:EventStruct. loc(E) |E|Label |
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event_msg_eq |
Def =msg=(E)(e_1,e_2) == (msg(E)(e_1)) =(MS(E)) (msg(E)(e_2)) |
| | Thm* E:EventStruct. =msg=(E) |E||E| |
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event_msg |
Def msg(E) == 1of(2of(2of(E))) |
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Thm* E:EventStruct. msg(E) |E||MS(E)| |
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event_msg_str |
Def MS(E) == 1of(2of(E)) |
| | Thm* E:EventStruct. MS(E) MessageStruct |
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swap_adjacent |
Def swap adjacent[P(x;y)](L1,L2)
== i:(||L1||-1). P(L1[i];L1[(i+1)]) & L2 = swap(L1;i;i+1) A List |
| | Thm* A:Type, P:(AAProp). swap adjacent[P(x,y)] (A List)(A List)Prop |
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swap |
Def swap(L;i;j) == (L o (i, j)) |
| | Thm* T:Type, L:T List, i,j:||L||. swap(L;i;j) T List |
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permute_list |
Def (L o f) == mklist(||L||;i.L[(f(i))]) |
| | Thm* T:Type, L:T List, f:(||L||||L||). (L o f) T List |
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select |
Def l[i] == hd(nth_tl(i;l)) |
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Thm* A:Type, l:A List, n:. 0n n < ||l|| l[n] A |
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lbl |
Def Label == {p:Pattern| ground_ptn(p) } |
| | Thm* Label Type |
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length |
Def ||as|| == Case of as; nil 0 ; a.as' ||as'||+1 (recursive) |
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Thm* A:Type, l:A List. ||l|| |
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Thm* ||nil|| |
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int_seg |
Def {i..j} == {k:| i k < j } |
| | Thm* m,n:. {m..n} Type |
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event_tag |
Def tag(E) == 1of(2of(2of(2of(2of(2of(E)))))) |
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Thm* E:TaggedEventStruct. tag(E) |E|Label |
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rel_star |
Def (R^*)(x,y) == n:. x R^n y |
| | Thm* T:Type, R:(TTProp). (R^*) TTProp |
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lelt |
Def i j < k == ij & j < k |
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nat |
Def == {i:| 0i } |
| | Thm* Type |
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le |
Def AB == B < A |
| | Thm* i,j:. (ij) Prop |
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not |
Def A == A False |
| | Thm* A:Prop. (A) Prop |
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assert |
Def b == if b True else False fi |
| | Thm* b:. b Prop |
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rel_or |
Def (R1 R2)(x,y) == (x R1 y) (x R2 y) |
| | Thm* T:Type, R1,R2:(TTProp). (R1 R2) TTProp |
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carrier |
Def |S| == 1of(S) |
| | Thm* S:Structure. |S| Type |
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msg_eq |
Def =(M)(m_1,m_2)
== ((content(M)(m_1)) =(cEQ(M)) (content(M)(m_2)))sender(M)(m_1) = sender(M)(m_2)
(uid(M)(m_1)=uid(M)(m_2)) |
| | Thm* M:MessageStruct. =(M) |M||M| |
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msg_id |
Def uid(MS) == 1of(2of(2of(2of(2of(MS))))) |
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Thm* M:MessageStruct. uid(M) |M| |
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msg_sender |
Def sender(MS) == 1of(2of(2of(2of(MS)))) |
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Thm* M:MessageStruct. sender(M) |M|Label |
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msg_content |
Def content(MS) == 1of(2of(2of(MS))) |
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Thm* M:MessageStruct. content(M) |M||cEQ(M)| |
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msg_content_eq |
Def cEQ(MS) == 1of(2of(MS)) |
| | Thm* M:MessageStruct. cEQ(M) DecidableEquiv |
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eq_dequiv |
Def =(DE) == 1of(2of(DE)) |
| | Thm* E:DecidableEquiv. =(E) |E||E| |
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pi2 |
Def 2of(t) == t.2 |
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Thm* A:Type, B:(AType), p:(a:AB(a)). 2of(p) B(1of(p)) |
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pi1 |
Def 1of(t) == t.1 |
| | Thm* A:Type, B:(AType), p:(a:AB(a)). 1of(p) A |
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nth_tl |
Def nth_tl(n;as) == if n0 as else nth_tl(n-1;tl(as)) fi (recursive) |
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Thm* A:Type, as:A List, i:. nth_tl(i;as) A List |
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ground_ptn |
Def ground_ptn(p)
== Case(p)
Case ptn_var(v) = >
false
Case ptn_pr( < x, y > ) = >
ground_ptn(x)ground_ptn(y)
Default = > true
(recursive) |
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Thm* p:Pattern. ground_ptn(p) |
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eq_lbl |
Def l1 = l2
== Case(l1)
Case ptn_atom(x) = >
Case(l2)
Case ptn_atom(y) = >
x=yAtom
Default = > false
Case ptn_int(x) = >
Case(l2)
Case ptn_int(y) = >
x=y
Default = > false
Case ptn_var(x) = >
Case(l2)
Case ptn_var(y) = >
x=yAtom
Default = > false
Case ptn_pr( < x, y > ) = >
Case(l2)
Case ptn_pr( < u, v > ) = >
x = uy = v
Default = > false
Default = > false
(recursive) |
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Thm* l1,l2:Pattern. l1 = l2 |
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case_ptn_var |
Def Case ptn_var(x) = > body(x) cont(x1,z)
== (x1.inr(x2) = >
(x1.inr(x2) = >
(x1.inl(x2) = > body(hd([x2 / tl(x1)])) cont(hd(x1),z))([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x1]) |
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case_ptn_int |
Def Case ptn_int(x) = > body(x) cont(x1,z)
== (x1.inr(x2) = >
(x1.inl(x2) = > body(hd([x2 / tl(x1)])) cont(hd(x1),z))([x2 / tl(x1)])
cont
(hd(x1)
,z))
([x1]) |
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hd |
Def hd(l) == Case of l; nil "?" ; h.t h |
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Thm* A:Type, l:A List. ||l||1 hd(l) A |
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Thm* A:Type, l:A List. hd(l) A |
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ptn |
Def Pattern == rec(T.ptn_con(T)) |
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Thm* Pattern Type |
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mklist |
Def mklist(n;f) == primrec(n;nil;i,l. l @ [(f(i))]) |
| | Thm* T:Type, n:, f:(nT). mklist(n;f) T List |
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append |
Def as @ bs == Case of as; nil bs ; a.as' [a / (as' @ bs)] (recursive) |
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Thm* T:Type, as,bs:T List. (as @ bs) T List |
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rel_exp |
Def R^n == if n=0 x,y. x = y T else x,y. z:T. (x R z) & (z R^n-1 y) fi
(recursive) |
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Thm* n:, T:Type, R:(TTProp). R^n TTProp |
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tl |
Def tl(l) == Case of l; nil nil ; h.t t |
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Thm* A:Type, l:A List. tl(l) A List |
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le_int |
Def ij == j < i |
| | Thm* i,j:. (ij) |
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case_default |
Def Default = > body(value,value) == body |
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band |
Def pq == if p q else false fi |
| | Thm* p,q:. (pq) |
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case_lbl_pair |
Def Case ptn_pr( < x, y > ) = > body(x;y) cont(x1,z)
== InjCase(x1; _. cont(z,z); x2.
InjCase(x2; _. cont(z,z); x2@0. InjCase(x2@0; _. cont(z,z); x2@1. x2@1/x3,x2@2. body(x3;x2@2)))) |
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case |
Def Case(value) body == body(value,value) |
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ptn_con |
Def ptn_con(T) == Atom++Atom+(TT) |
| | Thm* T:Type. ptn_con(T) Type |
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flip |
Def (i, j)(x) == if x=ij ;x=ji else x fi |
| | Thm* k:, i,j:k. (i, j) kk |
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primrec |
Def primrec(n;b;c) == if n=0 b else c(n-1,primrec(n-1;b;c)) fi (recursive) |
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Thm* T:Type, n:, b:T, c:(nTT). primrec(n;b;c) T |
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eq_int |
Def i=j == if i=j true ; false fi |
| | Thm* i,j:. (i=j) |
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lt_int |
Def i < j == if i < j true ; false fi |
| | Thm* i,j:. (i < j) |
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bnot |
Def b == if b false else true fi |
| | Thm* b:. b |
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case_inl |
Def inl(x) = > body(x) cont(value,contvalue)
== InjCase(value; x. body(x); _. cont(contvalue,contvalue)) |
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case_inr |
Def inr(x) = > body(x) cont(value,contvalue)
== InjCase(value; _. cont(contvalue,contvalue); x. body(x)) |
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eq_atom |
Def x=yAtom == if x=yAtomtrue; false fi |
| | Thm* x,y:Atom. x=yAtom |
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case_ptn_atom |
Def Case ptn_atom(x) = > body(x) cont(x1,z)
== InjCase(x1; x2. body(x2); _. cont(z,z)) |