{ 
[Info:Type]
es:EO+(Info)
[A:Type]
X:EClass(A)
[S:Type]
init:S. f:S 
A S. test:S A 
. nxt:S 
A S.
let Z = Threshold(init;f;test;nxt;X) in
e:E(Z)
(((
e 
prior(Z))
(
e':E(Z)
((e' = prior(Z)(e))
(e' <loc e)
(e'':E(X)
((e' <loc e'')
(e'' <loc e)
(
(test
list_accum(s,v.f s v;nxt Z(e');X(e', e''))
X(e'')))))
let s = list_accum(s,v.f s v;nxt Z(e');X(e', e)) in
((test s X(e))) (Z(e) = <s, X(e)>))))
((e prior(Z))
(e':E(X)
((e' <loc e)
((test list_accum(s,v.f s v;init;X(<e'))
X(e')))))
let s = list_accum(s,v.f s v;init;X(<e)) in
((test s X(e))) (Z(e) = <s, X(e)>))) }
{ Proof }
Definitions occuring in Statement :
es-threshold: Threshold(init;f;test;nxt;X),
es-prior-interface: prior(X),
es-prior-interval-vals: X(e1, e2),
es-prior-interface-vals: X(<e),
es-E-interface: E(X),
eclass-val: X(e),
in-eclass: e X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-locl: (e <loc e'),
es-E: E,
assert: b,
bool: ,
let: let,
uall: [x:A]. B[x],
all: x:A. B[x],
exists: x:A. B[x],
not: A,
implies: P Q,
or: P Q,
and: P Q,
apply: f a,
function: x:A B[x],
pair: <a, b>,
product: x:A B[x],
universe: Type,
equal: s = t,
list_accum: list_accum(x,a.f[x; a];y;l)
Definitions :
es-interface-at: X@i,
imax-class: (maximum f[v] 
lb with v from X),
es-filter-image: f[X],
map-class: (f[v] where v from X),
isl: isl(x),
can-apply: can-apply(f;x),
so_lambda: 
x.t[x],
fpf-cap: f(x)?z,
filter: filter(P;l),
record: record(x.T[x]),
list: type List,
true: True,
sq_type: SQType(T),
cond-class: [X?Y],
so_apply: x[s],
guard: {T},
eq_knd: a = b,
l_member: (x l),
fpf-dom: x dom(f),
limited-type: LimitedType,
bfalse: ff,
btrue: tt,
eq_bool: p =b q,
lt_int: i <z j,
le_int: i 
z j,
eq_int: (i =
j),
null: null(as),
set_blt: a < b,
grp_blt: a < b,
infix_ap: x f y,
dcdr-to-bool: [d],
bl-all: (xL.P[x])_b,
bl-exists: (xL.P[x])_b,
b-exists: (i<n.P[i])_b,
eq_type: eq_type(T;T'),
qeq: qeq(r;s),
q_less: q_less(r;s),
q_le: q_le(r;s),
deq-member: deq-member(eq;x;L),
deq-disjoint: deq-disjoint(eq;as;bs),
deq-all-disjoint: deq-all-disjoint(eq;ass;bs),
eq_str: Error :eq_str,
eq_id: a = b,
eq_lnk: a = b,
es-eq-E: e = e',
bimplies: p q,
band: p q,
bor: p q,
bnot: b,
int: 
,
unit: Unit,
intensional-universe: IType,
fpf: a:A fp-> B[a],
cand: A c B,
set: {x:A| B[x]} ,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
prop: 
,
atom: Atom,
es-base-E: es-base-E(es),
token: "$token",
record-select: r.x,
pair: <a, b>,
rev_implies: P 
Q,
es-prior-interface-vals: X(<e),
es-prior-interval-vals: X(e1, e2),
eclass-val: X(e),
list_accum: list_accum(x,a.f[x; a];y;l),
es-prior-interface: prior(X),
ifthenelse: if b then t else f fi ,
iff: P Q,
subtype: S 
T,
lambda: x.A[x],
member: t T,
strong-subtype: strong-subtype(A;B),
eq_atom: x =a y,
eq_atom: eq_atom$n(x;y),
dep-isect: Error :dep-isect,
record+: record+,
le: A B,
ge: i j ,
less_than: a < b,
uimplies: b supposing a,
uiff: uiff(P;Q),
subtype_rel: A r B,
es-threshold: Threshold(init;f;test;nxt;X),
es-E: E,
top: Top,
es-E-interface: E(X),
event_ordering: EO,
es-locl: (e <loc e'),
void: Void,
equal: s = t,
exists: x:A. B[x],
and: P Q,
union: left + right,
false: False,
or: P Q,
implies: P Q,
not: A,
product: x:A B[x],
bool: ,
event-ordering+: EO+(Info),
eclass: EClass(A[eo; e]),
so_lambda: x y.t[x; y],
uall: [x:A]. B[x],
isect: 
x:A. B[x],
let: let,
apply: f a,
universe: Type,
all: x:A. B[x],
function: x:A B[x],
squash: 
T,
sqequal: s ~ t,
MaAuto: Error :MaAuto,
Complete: Error :Complete,
Try: Error :Try,
THENL_cons: Error :THENL_nil,
THENL_cons: Error :THENL_cons,
SplitOn: Error :SplitOn,
CollapseTHENA: Error :CollapseTHENA,
THENL_v2: Error :THENL,
RepUR: Error :RepUR,
CollapseTHEN: Error :CollapseTHEN,
Auto: Error :Auto,
in-eclass: e X,
assert: b,
AssertBY: Error :AssertBY,
THENM: Error :THENM,
tactic: Error :tactic,
pi1: fst(t),
es-loc: loc(e),
Id: Id,
es-causl: (e < e'),
es-le: e loc e'
Lemmas :
es-causl_wf,
Id_wf,
es-locl_transitivity2,
es-le_weakening,
es-causl_weakening,
es-is-prior-interface,
set_subtype_base,
es-prior-interface-val-unique2,
es-prior-interface-val,
assert_witness,
squash_wf,
bool_wf,
es-E-interface_wf,
es-threshold_wf,
not_wf,
assert_wf,
es-locl_wf,
es-E_wf,
eclass_wf,
event-ordering+_wf,
event-ordering+_inc,
is-threshold,
iff_wf,
es-base-E_wf,
subtype_rel_self,
in-eclass_wf,
member_wf,
es-interface-top,
subtype_rel_wf,
intensional-universe_wf,
iff_weakening_uiff,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
bnot_wf,
uiff_inversion,
es-prior-interface_wf1,
es-prior-interface_wf,
es-interface-subtype_rel2,
top_wf,
rev_implies_wf,
true_wf,
false_wf,
unit_wf,
eclass-val_wf,
es-prior-interval-vals_wf,
list_accum_wf,
es-prior-interface-vals_wf,
ifthenelse_wf,
list-subtype,
l_member_wf,
eclass-val_wf2,
es-interface-val_wf2,
es-threshold-val,
uall_wf,
assert_elim,
es-interface-val_wf,
subtype_base_sq,
bool_subtype_base,
bool_cases
\mforall{}[Info:Type]
\mforall{}es:EO+(Info)
\mforall{}[A:Type]
\mforall{}X:EClass(A)
\mforall{}[S:Type]
\mforall{}init:S. \mforall{}f:S {}\mrightarrow{} A {}\mrightarrow{} S. \mforall{}test:S {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}. \mforall{}nxt:S \mtimes{} A {}\mrightarrow{} S.
let Z = Threshold(init;f;test;nxt;X) in
\mforall{}e:E(Z)
(((\muparrow{}e \mmember{}\msubb{} prior(Z))
\mwedge{} (\mexists{}e':E(Z)
((e' = prior(Z)(e))
\mwedge{} (e' <loc e)
\mwedge{} (\mforall{}e'':E(X)
((e' <loc e'')
{}\mRightarrow{} (e'' <loc e)
{}\mRightarrow{} (\mneg{}\muparrow{}(test list\_accum(s,v.f s v;nxt Z(e');X(e', e'')) X(e'')))))
\mwedge{} let s = list\_accum(s,v.f s v;nxt Z(e');X(e', e)) in
(\muparrow{}(test s X(e))) \mwedge{} (Z(e) = <s, X(e)>))))
\mvee{} ((\mneg{}\muparrow{}e \mmember{}\msubb{} prior(Z))
\mwedge{} (\mforall{}e':E(X)
((e' <loc e) {}\mRightarrow{} (\mneg{}\muparrow{}(test list\_accum(s,v.f s v;init;X(<e')) X(e')))))
\mwedge{} let s = list\_accum(s,v.f s v;init;X(<e)) in
(\muparrow{}(test s X(e))) \mwedge{} (Z(e) = <s, X(e)>)))
Date html generated:
2011_08_16-PM-05_12_08
Last ObjectModification:
2011_06_20-AM-01_13_44
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