Nuprl Lemma : reliable-env-property
∀[M:Type ⟶ Type]
∀S0:InitialSystem(P.M[P]). ∀n2m:ℕ ⟶ pMsg(P.M[P]). ∀l2m:Id ⟶ pMsg(P.M[P]). ∀env:pEnvType(P.M[P]).
let r = pRun(S0;env;n2m;l2m) in
reliable-env(env; r)
⇒ (∀tn:run-msg-commands(r)
∃e:runEvents(r)
let t,n = tn
in (run-info(r;e)
= intransit-to-info(n2m;l2m;r;env;run-event-step(e);run-command(r;t;n))
∈ (ℤ × Id × pMsg(P.M[P])))
∧ (run-event-loc(e) = (fst(snd(run-command(r;t;n)))) ∈ Id))
supposing Continuous+(P.M[P])
Proof
Definitions occuring in Statement :
intransit-to-info: intransit-to-info(n2m;l2m;r;env;t;lbl),
run-msg-commands: run-msg-commands(r),
run-command: run-command(r;t;n),
reliable-env: reliable-env(env; r),
InitialSystem: InitialSystem(P.M[P]),
run-event-step: run-event-step(e),
run-event-loc: run-event-loc(e),
runEvents: runEvents(r),
run-info: run-info(r;e),
pRun: pRun(S0;env;nat2msg;loc2msg),
pEnvType: pEnvType(T.M[T]),
pMsg: pMsg(P.M[P]),
Id: Id,
strong-type-continuous: Continuous+(T.F[T]),
nat: ℕ,
let: let,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
spread: spread def,
product: x:A × B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
strong-type-continuous: Continuous+(T.F[T]),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
let: let,
so_lambda: λ2x.t[x],
so_apply: x[s],
InitialSystem: InitialSystem(P.M[P]),
implies: P ⇒ Q,
run-msg-commands: run-msg-commands(r),
cand: A c∧ B,
run-command-node: run-command-node(r;t;n),
prop: ℙ,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
pi1: fst(t),
pi2: snd(t),
sq_type: SQType(T),
guard: {T},
int_seg: {i..j-},
exists: ∃x:A. B[x],
pInTransit: pInTransit(P.M[P]),
sq_stable: SqStable(P),
squash: ↓T,
ldag: LabeledDAG(T),
pRunType: pRunType(T.M[T]),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
run-command: run-command(r;t;n),
pRun: pRun(S0;env;nat2msg;loc2msg),
ycomb: Y,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
run-intransit: run-intransit(r;t),
run-system: run-system(r;t),
fulpRunType: fulpRunType(T.M[T]),
System: System(P.M[P]),
pEnvType: pEnvType(T.M[T]),
nat_plus: ℕ+,
spreadn: spread3,
lelt: i ≤ j < k,
less_than: a < b,
do-chosen-command: do-chosen-command(nat2msg;loc2msg;t;S;n;m;nm),
lg-is-source: lg-is-source(g;i),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
create-component: create-component(t;P;x;Cs;L),
int_upper: {i...},
runEvents: runEvents(r),
is-run-event: is-run-event(r;t;x),
isl: isl(x),
outl: outl(x),
band: p ∧b q,
rev_uimplies: rev_uimplies(P;Q),
eq_atom: x =a y,
run-event-loc: run-event-loc(e),
run-event-step: run-event-step(e),
intransit-to-info: intransit-to-info(n2m;l2m;r;env;t;lbl),
run-info: run-info(r;e),
command-to-msg: command-to-msg(c;nmsg;lmsg)
Latex:
\mforall{}[M:Type {}\mrightarrow{} Type]
\mforall{}S0:InitialSystem(P.M[P]). \mforall{}n2m:\mBbbN{} {}\mrightarrow{} pMsg(P.M[P]). \mforall{}l2m:Id {}\mrightarrow{} pMsg(P.M[P]). \mforall{}env:pEnvType(P.M[P]).
let r = pRun(S0;env;n2m;l2m) in
reliable-env(env; r)
{}\mRightarrow{} (\mforall{}tn:run-msg-commands(r)
\mexists{}e:runEvents(r)
let t,n = tn
in (run-info(r;e)
= intransit-to-info(n2m;l2m;r;env;run-event-step(e);run-command(r;t;n)))
\mwedge{} (run-event-loc(e) = (fst(snd(run-command(r;t;n))))))
supposing Continuous+(P.M[P])
Date html generated:
2016_05_17-AM-11_00_10
Last ObjectModification:
2016_01_18-AM-00_27_00
Theory : process-model
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