Nuprl Lemma : reliable-env-property2
∀[M:Type ─→ Type]
∀S:InitialSystem(P.M[P]). ∀n2m:ℕ ─→ pMsg(P.M[P]). ∀l2m:Id ─→ pMsg(P.M[P]). ∀env:pEnvType(P.M[P]).
let r = pRun(S;env;n2m;l2m) in
reliable-env(env; r)
⇒ (∀e:runEvents(r). ∀P:Process(P.M[P]).
((P ∈ run-event-state-when(r;e))
⇒ ∀p∈snd((P (snd(run-info(r;e))))).let y,c = p
in (com-kind(c) ∈ ``msg choose new``)
⇒ (∃e':runEvents(r)
((run-event-loc(e') = y ∈ Id)
∧ (e run-lt(r) e')
∧ (∃n:ℕ
∃nm:Id
((snd(run-info(r;e')))
= command-to-msg(c;n2m n;l2m nm)
∈ pMsg(P.M[P])))
∧ ((run-cause(r) e') = (inl e) ∈ (runEvents(r)?))))))
supposing Continuous+(P.M[P])
Proof
Definitions occuring in Statement :
command-to-msg: command-to-msg(c;nmsg;lmsg),
reliable-env: reliable-env(env; r),
InitialSystem: InitialSystem(P.M[P]),
run-lt: run-lt(r),
run-cause: run-cause(r),
run-event-loc: run-event-loc(e),
run-event-state-when: run-event-state-when(r;e),
runEvents: runEvents(r),
run-info: run-info(r;e),
pRun: pRun(S0;env;nat2msg;loc2msg),
pEnvType: pEnvType(T.M[T]),
com-kind: com-kind(c),
pMsg: pMsg(P.M[P]),
Process: Process(P.M[P]),
lg-all: ∀x∈G.P[x],
Id: Id,
l_member: (x ∈ l),
cons: [a / b],
nil: [],
strong-type-continuous: Continuous+(T.F[T]),
nat: ℕ,
let: let,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ─→ B[x],
spread: spread def,
inl: inl x,
union: left + right,
token: "$token",
atom: Atom,
universe: Type,
equal: s = t ∈ T
Lemmas :
reliable-env-property,
nat_wf,
pRun_wf2,
run-event-step-positive,
l_member_wf,
Process_wf,
run-event-state-when_wf,
pRun_wf,
less_than_wf,
runEvents_wf,
reliable-env_wf,
pEnvType_wf,
Id_wf,
pMsg_wf,
InitialSystem_wf,
strong-type-continuous_wf,
run-info_wf,
Process-apply_wf,
pExt_wf,
lg-label_wf,
pCom_wf,
com-kind_wf,
cons_wf,
nil_wf,
int_seg_wf,
lg-size_wf,
sq_stable__assert,
is-run-event_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
false_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
fulpRunType_wf,
subtract_wf,
decidable__le,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
unit_wf2,
System_wf,
subtype_rel_dep_function,
top_wf,
ldag_wf,
pInTransit_wf,
int_seg_subtype-nat,
subtype_rel_self,
lg-is-source_wf,
eq_id_wf,
component_wf,
lelt_wf,
eq_atom_wf,
assert_of_eq_atom,
comm-msg_wf,
neg_assert_of_eq_atom,
mapfilter_wf,
assert_wf,
lt_int_wf,
bnot_wf,
not_wf,
assert_elim,
bfalse_wf,
btrue_neq_bfalse,
bool_cases,
assert_of_lt_int,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert-eq-id,
and_wf,
pi2_wf,
member-mapfilter,
set_wf,
squash_wf,
true_wf,
lg-contains_wf,
add-cause_wf,
deliver-msg_wf,
lg-remove_wf_dag,
atom2_subtype_base,
pi1_wf_top,
subtype_rel_product,
subtype_top,
length_wf,
list_wf,
int_upper_wf,
list_induction,
all_wf,
select_wf,
sq_stable__le,
list_accum_wf,
deliver-msg-to-comp_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
list_accum_nil_lemma,
less_than_transitivity1,
less_than_irreflexivity,
length_of_cons_lemma,
list_accum_cons_lemma,
le_int_wf,
assert_of_le_int,
minus-zero,
decidable__lt,
select-cons,
lg-contains_transitivity,
is-dag_wf,
lg-append_wf,
lg-append_wf_dag,
lg-append-contains,
lg-contains_weakening,
lg-label-map,
subtype_rel-labeled-graph,
length-map-sq,
subtype_rel_set,
labeled-graph_wf,
dep-isect-subtype,
zero-le-nat,
lg-size-append,
add_functionality_wrt_eq,
iff_weakening_equal,
add-mul-special,
zero-mul,
lg-label-append,
exists_wf,
lg-map_wf,
lg-size-map,
pRunType_wf,
intransit-to-info_wf,
run-event-step_wf,
rel-rel-plus,
run-pred_wf,
command-to-msg_wf,
not_assert_elim,
assert_of_band,
it_wf,
run-event-loc_wf,
run-lt_wf,
assert_functionality_wrt_uiff,
subtype_rel_sets,
run-cause_wf
Latex:
\mforall{}[M:Type {}\mrightarrow{} Type]
\mforall{}S: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(S;env;n2m;l2m) in
reliable-env(env; r)
{}\mRightarrow{} (\mforall{}e:runEvents(r). \mforall{}P:Process(P.M[P]).
((P \mmember{} run-event-state-when(r;e))
{}\mRightarrow{} \mforall{}p\mmember{}snd((P (snd(run-info(r;e))))).let y,c = p
in (com-kind(c) \mmember{} ``msg choose new``)
{}\mRightarrow{} (\mexists{}e':runEvents(r)
((run-event-loc(e') = y)
\mwedge{} (e run-lt(r) e')
\mwedge{} (\mexists{}n:\mBbbN{}
\mexists{}nm:Id
((snd(run-info(r;e')))
= command-to-msg(c;n2m n;l2m nm)))
\mwedge{} ((run-cause(r) e') = (inl e))))))
supposing Continuous+(P.M[P])
Date html generated:
2015_07_23-AM-11_19_24
Last ObjectModification:
2015_02_04-PM-04_56_22
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