Nuprl Lemma : local-simulation-validity
∀correct:Id ─→ ℙ. ∀g,f:Name ─→ Type. ∀X:EClass(Interface).
(LocalClass(X)
⇒ (∀locs:bag(Id). ∀hdr:Name.
∀hdrs:Name List. ∀es:EO+(Message(f)).
((∀i:Id. ((correct i) ⇒ local-simulation-input-validity(g;X;hdr;locs;hdrs;es;i)))
⇒ (∀P:Interface ─→ ℙ. ∀R:Interface ─→ Message(g) ─→ ℙ.
((∀eo:EO+(Message(g))
(eo-msg-interface-constraint(eo;X;hdrs;g)
⇒ (∀e:E. ∀v:Interface. (v ∈ X(e) ⇒ P[v] ⇒ (↓∃e':E. ((e' < e) ∧ R[v;info(e')]))))))
⇒ (∀e:E. ∀v:Interface.
((correct loc(e))
⇒ v ∈ local-simulation-class(X;locs;hdr)(e)
⇒ P[v]
⇒ (↓∃e':E
((e' <loc e)
∧ (↑has-header-and-in-locs(info(e');hdr;locs))
∧ R[v;snd(msg-body(info(e')))])))))))
supposing hdr encodes Id × Message(g)))
Proof
Definitions occuring in Statement :
local-simulation-input-validity: local-simulation-input-validity(g;X;hdr;locs;hdrs;es;i),
eo-msg-interface-constraint: eo-msg-interface-constraint(es;X;hdrs;f),
msg-interface: Interface,
has-header-and-in-locs: has-header-and-in-locs(msg;hdr;locs),
local-simulation-class: local-simulation-class(X;locs;hdr),
encodes-msg-type: hdr encodes T,
msg-body: msg-body(msg),
Message: Message(f),
local-class: LocalClass(X),
classrel: v ∈ X(e),
eclass: EClass(A[eo; e]),
es-info: info(e),
event-ordering+: EO+(Info),
es-locl: (e <loc e'),
es-causl: (e < e'),
es-loc: loc(e),
es-E: E,
Id: Id,
name: Name,
list: T List,
assert: ↑b,
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
product: x:A × B[x],
universe: Type,
bag: bag(T)
Lemmas :
global-eo-causl,
local-simulation-inputs_wf,
Message_wf,
subtype_rel_list,
Id_wf,
bag-member_wf,
top_wf,
local-simulation-event_wf,
es-E_wf,
local-simulation-eo_wf,
event-ordering+_subtype,
global-eo-E-sq,
filter_wf5,
es-before_wf,
compose_wf,
l_member_wf,
bool_wf,
has-header-and-in-locs_wf,
subtype_rel_product,
subtype_top,
subtype_rel_transitivity,
es-info_wf,
filter-map,
length-map,
select_member,
lelt_wf,
length_wf,
member_filter,
select_wf,
sq_stable__le,
member-es-before,
es-locl_wf,
assert_wf,
global-eo-info,
filter_cons_lemma,
filter_nil_lemma,
filter_append_sq,
map_append_sq,
filter_type,
name_wf,
msg-header_wf,
assert-has-header-and-in-locs,
list_wf,
less_than_wf,
map_wf,
eqtt_to_assert,
cons_wf,
nil_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
le_wf,
lt_int_wf,
assert_of_lt_int,
select-map,
assert-bnot,
select-append,
map-map,
set_wf,
msg-body_wf2,
subtype_rel-equal,
msg-type_wf,
iff_weakening_equal
Latex:
\mforall{}correct:Id {}\mrightarrow{} \mBbbP{}. \mforall{}g,f:Name {}\mrightarrow{} Type. \mforall{}X:EClass(Interface).
(LocalClass(X)
{}\mRightarrow{} (\mforall{}locs:bag(Id). \mforall{}hdr:Name.
\mforall{}hdrs:Name List. \mforall{}es:EO+(Message(f)).
((\mforall{}i:Id. ((correct i) {}\mRightarrow{} local-simulation-input-validity(g;X;hdr;locs;hdrs;es;i)))
{}\mRightarrow{} (\mforall{}P:Interface {}\mrightarrow{} \mBbbP{}. \mforall{}R:Interface {}\mrightarrow{} Message(g) {}\mrightarrow{} \mBbbP{}.
((\mforall{}eo:EO+(Message(g))
(eo-msg-interface-constraint(eo;X;hdrs;g)
{}\mRightarrow{} (\mforall{}e:E. \mforall{}v:Interface.
(v \mmember{} X(e) {}\mRightarrow{} P[v] {}\mRightarrow{} (\mdownarrow{}\mexists{}e':E. ((e' < e) \mwedge{} R[v;info(e')]))))))
{}\mRightarrow{} (\mforall{}e:E. \mforall{}v:Interface.
((correct loc(e))
{}\mRightarrow{} v \mmember{} local-simulation-class(X;locs;hdr)(e)
{}\mRightarrow{} P[v]
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((e' <loc e)
\mwedge{} (\muparrow{}has-header-and-in-locs(info(e');hdr;locs))
\mwedge{} R[v;snd(msg-body(info(e')))])))))))
supposing hdr encodes Id \mtimes{} Message(g)))
Date html generated:
2015_07_22-PM-00_07_15
Last ObjectModification:
2015_02_04-PM-04_43_16
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