{ 
[Info:Type]. [es:EO+(Info)]. [A,B:Type]. [X:EClass(A)]. [P:B 
].
[num:A 
]. [init:B]. [f:B A B]. [e:E].
es-collect-accum(X;x.num[x];init;b,v.f[b;v];b.P[b])(e)
= <num[X(e)]
, list_accum(b,v.f[b;v];
init;
mapfilter(
e.X(e);e'.(num[X(e')] =
num[X(e)]);
(X)(e)))
>
supposing 
e 
es-collect-accum(X;x.num[x];init;b,v.f[b;v];b.P[b]) }
{ Proof }
Definitions occuring in Statement :
es-collect-accum: es-collect-accum(X;x.num[x];init;a,v.f[a; v];a.P[a]),
es-interface-predecessors: (X)(e),
eclass-val: X(e),
in-eclass: e X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
eq_int: (i = j),
assert: b,
bool: ,
nat: ,
uimplies: b supposing a,
uall: [x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
lambda: x.A[x],
function: x:A B[x],
pair: <a, b>,
product: x:A 
B[x],
universe: Type,
equal: s = t,
mapfilter: mapfilter(f;P;L),
list_accum: list_accum(x,a.f[x; a];y;l)
Definitions :
eq_id: a = b,
es-prior-interface: prior(X),
exists: 
x:A. B[x],
es-interface-at: X@i,
tag-by: zT,
fset: FSet{T},
dataflow: dataflow(A;B),
isect2: T1 
T2,
b-union: A 
B,
fpf-cap: f(x)?z,
record: record(x.T[x]),
cand: A c
B,
is_list_splitting: is_list_splitting(T;L;LL;L2;f),
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
req: x = y,
rnonneg: rnonneg(r),
rleq: x y,
i-member: r I,
partitions: partitions(I;p),
modulus-of-ccontinuity: modulus-of-ccontinuity(omega;I;f),
fpf-sub: f 
g,
sq_stable: SqStable(P),
bag-only: only(bs),
bag_only_single: bag_only_single{bag_only_single_compseq_tag_def:o}(x),
es-filter-image: f[X],
rev_uimplies: rev_uimplies(P;Q),
length: ||as||,
eq_knd: a = b,
fpf-dom: x dom(f),
intensional-universe: IType,
bfalse: ff,
unit: Unit,
int_eq: if a=b then c else d,
btrue: tt,
atom_eq: atomeqn def,
sq_type: SQType(T),
sqequal: s ~ t,
union: left + right,
or: P 
Q,
append: as @ bs,
guard: {T},
locl: locl(a),
Knd: Knd,
atom: Atom$n,
filter: filter(P;l),
l_member: (x l),
list: type List,
natural_number: $n,
rationals: 
,
real: 
,
grp_car: |g|,
int: 
,
es-loc: loc(e),
label: ...$L... t,
true: True,
squash: 
T,
rev_implies: P 
Q,
implies: P 
Q,
iff: P Q,
Id: Id,
fpf: a:A fp-> B[a],
strong-subtype: strong-subtype(A;B),
false: False,
collect-event: collect-event(es;X;n;v.num[v];L.P[L];e),
le: A B,
ge: i 
j ,
not: 
A,
less_than: a < b,
es-first-at: e is first@ i s.t. e.P[e],
alle-lt: e<e'.P[e],
and: P Q,
uiff: uiff(P;Q),
void: Void,
subtype: S T,
atom: Atom,
es-base-E: es-base-E(es),
token: "$token",
es-E-interface: E(X),
subtype_rel: A r B,
quotient: x,y:A//B[x; y],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
ifthenelse: if b then t else f fi ,
top: Top,
all: x:A. B[x],
in-eclass: e X,
dep-isect: Error :dep-isect,
eq_atom: x =a y,
eq_atom: eq_atom$n(x;y),
record+: record+,
bag: bag(T),
set: {x:A| B[x]} ,
record-select: r.x,
axiom: Ax,
es-interface-predecessors: (X)(e),
eq_int: (i = j),
mapfilter: mapfilter(f;P;L),
apply: f a,
es-collect-accum: es-collect-accum(X;x.num[x];init;a,v.f[a; v];a.P[a]),
eclass-val: X(e),
product: x:A B[x],
prop: 
,
assert: b,
event_ordering: EO,
es-E: E,
nat: ,
uimplies: b supposing a,
equal: s = t,
event-ordering+: EO+(Info),
universe: Type,
so_lambda: 
x y.t[x; y],
eclass: EClass(A[eo; e]),
uall: [x:A]. B[x],
isect: x:A. B[x],
member: t T,
function: x:A B[x],
bool: ,
MaAuto: Error :MaAuto,
CollapseTHEN: Error :CollapseTHEN,
THENM: Error :THENM,
CollapseTHENA: Error :CollapseTHENA,
Auto: Error :Auto,
so_apply: x[s1;s2],
list_accum: list_accum(x,a.f[x; a];y;l),
so_apply: x[s],
es-collect: Collect(X;x.num[x];L.P[L]),
pi2: snd(t),
pi1: fst(t),
pair: <a, b>,
single-bag: {x},
lambda: x.A[x],
so_lambda: x.t[x]
Lemmas :
event-ordering+_wf,
event-ordering+_inc,
subtype_rel_self,
es-base-E_wf,
es-E_wf,
top_wf,
nat_wf,
subtype_rel_wf,
es-collect-accum_wf,
es-interface-subtype_rel2,
es-interface-top,
member_wf,
eclass_wf,
in-eclass_wf,
assert_wf,
bool_wf,
is-collect-accum,
eclass-val_wf,
es-interface-predecessors_wf,
Id_wf,
es-E-interface_wf,
mapfilter_wf,
list_accum_wf,
iff_wf,
rev_implies_wf,
squash_wf,
es-collect-accum-es-collect,
es-loc_wf,
eq_int_wf,
false_wf,
ifthenelse_wf,
true_wf,
le_wf,
not_wf,
list-subtype,
l_member_wf,
uiff_wf,
assert-eq-id,
subtype_base_sq,
bool_subtype_base,
assert_elim,
btrue_wf,
bfalse_wf,
unit_wf,
intensional-universe_wf,
es-interface-val_wf2,
es-collect_wf,
length_wf1,
es-filter-image-val,
is-es-collect,
es-collect-val,
pi1_wf_top,
pi2_wf,
sq_stable__assert
\mforall{}[Info:Type]. \mforall{}[es:EO+(Info)]. \mforall{}[A,B:Type]. \mforall{}[X:EClass(A)]. \mforall{}[P:B {}\mrightarrow{} \mBbbB{}]. \mforall{}[num:A {}\mrightarrow{} \mBbbN{}]. \mforall{}[init:B].
\mforall{}[f:B {}\mrightarrow{} A {}\mrightarrow{} B]. \mforall{}[e:E].
es-collect-accum(X;x.num[x];init;b,v.f[b;v];b.P[b])(e)
= <num[X(e)]
, list\_accum(b,v.f[b;v];init;mapfilter(\mlambda{}e.X(e);\mlambda{}e'.(num[X(e')] =\msubz{} num[X(e)]);\mleq{}(X)(e)))
>
supposing \muparrow{}e \mmember{}\msubb{} es-collect-accum(X;x.num[x];init;b,v.f[b;v];b.P[b])
Date html generated:
2011_08_16-PM-05_28_22
Last ObjectModification:
2011_06_20-AM-01_24_15
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