Nuprl Lemma : CLK-ilf
∀[MsgType:{T:Type| valueall-type(T)} ]. ∀[locs:bag(Id)]. ∀[reply:Id ⟶ MsgType ⟶ (MsgType × Id)].
∀[f:CLK_headers_type{i:l}(MsgType)]. ∀[es:EO+(Message(f))]. ∀[e:E]. ∀[d:ℤ]. ∀[i:Id]. ∀[m:Message(f)].
{<d, i, m> ∈ CLK_main(MsgType;locs;reply;f)(e)
⇐⇒ loc(e) ↓∈ locs
∧ ((header(e) = ``CLK msg`` ∈ Name) ∧ has-es-info-type(es;e;f;MsgType × ℤ))
∧ (d = 0 ∈ ℤ)
∧ (i = (snd((reply loc(e) (fst(msgval(e)))))) ∈ Id)
∧ (m = make-Msg(``CLK msg``;<fst((reply loc(e) (fst(msgval(e))))), CLK_ClockVal(MsgType;f)@e>) ∈ Message(f))}
Proof
Definitions occuring in Statement :
CLK_main: CLK_main(MsgType;locs;reply;f),
CLK_ClockFun: CLK_ClockVal(MsgType;f)@e,
CLK_headers_type: CLK_headers_type{i:l}(MsgType),
msg-interface: Interface,
make-Msg: make-Msg(hdr;val),
es-info-body: msgval(e),
has-es-info-type: has-es-info-type(es;e;f;T),
es-header: header(e),
Message: Message(f),
classrel: v ∈ X(e),
event-ordering+: EO+(Info),
es-loc: loc(e),
es-E: E,
Id: Id,
name: Name,
cons: [a / b],
nil: [],
valueall-type: valueall-type(T),
uall: ∀[x:A]. B[x],
guard: {T},
pi1: fst(t),
pi2: snd(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
token: "$token",
universe: Type,
equal: s = t ∈ T,
bag-member: x ↓∈ bs,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
CLK_headers_type: CLK_headers_type{i:l}(MsgType),
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
vatype: ValueAllType,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
listp: A List+,
name: Name,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T,
length: ||as||,
list_ind: list_ind,
CLK_headers: CLK_headers(),
cons: [a / b],
nil: [],
it: ⋅,
true: True,
select: L[n],
uimplies: b supposing a,
guard: {T},
rev_implies: P ⇐ Q,
CLK_headers_fun: CLK_headers_fun(MsgType),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
btrue: tt,
null: null(as),
bag-member: x ↓∈ bs,
has-es-info-type: has-es-info-type(es;e;f;T),
make-msg-interface: make-msg-interface(i;l;m),
classrel: v ∈ X(e),
CLK_main: CLK_main(MsgType;locs;reply;f),
CLK_Reply: CLK_Reply(MsgType;reply;f),
equal-info-body: v = body(e),
CLK_msg'base: CLK_msg'base(MsgType;f),
encodes-msg-type: hdr encodes T,
uiff: uiff(P;Q),
cand: A c∧ B,
CLK_mk_reply: CLK_mk_reply(MsgType;reply;f),
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
pi1: fst(t),
exists: ∃x:A. B[x],
msg-interface: Interface,
CLK_msg'send: CLK_msg'send(MsgType;f),
mk-msg-interface: mk-msg-interface(l;m),
pi2: snd(t),
sq_stable: SqStable(P)
Latex:
\mforall{}[MsgType:\{T:Type| valueall-type(T)\} ]. \mforall{}[locs:bag(Id)]. \mforall{}[reply:Id {}\mrightarrow{} MsgType {}\mrightarrow{} (MsgType \mtimes{} Id)].
\mforall{}[f:CLK\_headers\_type\{i:l\}(MsgType)]. \mforall{}[es:EO+(Message(f))]. \mforall{}[e:E]. \mforall{}[d:\mBbbZ{}]. \mforall{}[i:Id].
\mforall{}[m:Message(f)].
\{<d, i, m> \mmember{} CLK\_main(MsgType;locs;reply;f)(e)
\mLeftarrow{}{}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} locs
\mwedge{} ((header(e) = ``CLK msg``) \mwedge{} has-es-info-type(es;e;f;MsgType \mtimes{} \mBbbZ{}))
\mwedge{} (d = 0)
\mwedge{} (i = (snd((reply loc(e) (fst(msgval(e)))))))
\mwedge{} (m = make-Msg(``CLK msg``;<fst((reply loc(e) (fst(msgval(e))))), CLK\_ClockVal(MsgType;f)@e>)\000C)\}
Date html generated:
2016_05_17-PM-02_45_46
Last ObjectModification:
2016_04_03-PM-09_42_55
Theory : lamport!s!clocks
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