{ 
[A:
']. [dfps:DataflowProgram(A) List].
[B:{B:Type| valueall-type(B)} ].
[F:k:
||dfps|| 
bag(df-program-type(dfps[k])) bag(B)].
(parallel-df-program(B;F;dfps) 
DataflowProgram(A))
supposing 0 < ||dfps|| }
{ Proof }
Definitions occuring in Statement :
parallel-df-program: parallel-df-program(B;F;dfps),
df-program-type: df-program-type(dfp),
dataflow-program: DataflowProgram(A),
select: l[i],
length: ||as||,
int_seg: {i..j
},
uimplies: b supposing a,
uall: [x:A]. B[x],
member: t T,
less_than: a < b,
set: {x:A| B[x]} ,
function: x:A B[x],
list: type List,
natural_number: $n,
universe: Type,
bag: bag(T),
valueall-type: valueall-type(T)
Definitions :
list_ind: list_ind def,
permutation: permutation(T;L1;L2),
quotient: x,y:A//B[x; y],
l_member: (x l),
parallel-df-prog1: parallel-df-prog1(B;G;dfp),
lambda: 
x.A[x],
parallel-df-prog2: parallel-df-prog2(B;G;dfp1;dfp2),
ifthenelse: if b then t else f fi ,
tl: tl(l),
hd: hd(l),
bfalse: ff,
limited-type: LimitedType,
btrue: tt,
le_int: i 
z j,
eq_int: (i =
j),
eq_atom: x =a y,
null: null(as),
set_blt: a <
b,
grp_blt: a < b,
apply: f a,
infix_ap: x f y,
dcdr-to-bool: [d],
bl-all: (xL.P[x])_b,
bl-exists: (
xL.P[x])_b,
b-exists: (i<n.P[i])_b,
eq_type: eq_type(T;T'),
eq_atom: eq_atom$n(x;y),
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),
name_eq: name_eq(x;y),
eq_id: a = b,
eq_lnk: a = b,
bimplies: p 
q,
band: p 
q,
bor: p 
q,
lt_int: i <z j,
assert: 
b,
bnot: 
b,
unit: Unit,
union: left + right,
bool: 
,
fpf: a:A fp-> B[a],
strong-subtype: strong-subtype(A;B),
ge: i 
j ,
uiff: uiff(P;Q),
subtype_rel: A 
r B,
top: Top,
product: x:A 
B[x],
grp_car: |g|,
nat: ,
guard: {T},
and: P Q,
lelt: i j < k,
void: Void,
implies: P Q,
false: False,
not: A,
le: A B,
real: 
,
rationals: 
,
subtype: S T,
int: 
,
all: x:A. B[x],
length: ||as||,
natural_number: $n,
axiom: Ax,
parallel-df-program: parallel-df-program(B;F;dfps),
equal: s = t,
prop: 
,
function: x:A B[x],
bag: bag(T),
df-program-type: df-program-type(dfp),
select: l[i],
int_seg: {i..j},
set: {x:A| B[x]} ,
valueall-type: valueall-type(T),
less_than: a < b,
list: type List,
universe: Type,
uimplies: b supposing a,
isect: 
x:A. B[x],
uall: [x:A]. B[x],
so_lambda: 
x.t[x],
member: t T,
dataflow-program: DataflowProgram(A),
minus: -n,
add: n + m,
subtract: n - m,
Repeat: Error :Repeat,
CollapseTHEN: Error :CollapseTHEN,
Auto: Error :Auto,
D: Error :D,
CollapseTHENA: Error :CollapseTHENA,
Complete: Error :Complete,
Try: Error :Try,
proper-iseg: L1 < L2,
iseg: l1 l2,
gt: i > j,
map: map(f;as),
sqequal: s ~ t,
tag-by: zT,
rev_implies: P 
Q,
iff: P Q,
record+: record+,
record: record(x.T[x]),
fset: FSet{T},
isect2: T1 T2,
stream: stream(A),
dataflow: dataflow(A;B),
intensional-universe: IType,
fpf-sub: f g,
deq: EqDecider(T),
ma-state: State(ds),
true: True,
squash: 
T,
fpf-cap: f(x)?z,
refl: Refl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
so_lambda: x y.t[x; y],
sq_type: SQType(T),
IdLnk: IdLnk,
Id: Id,
append: as @ bs,
locl: locl(a),
Knd: Knd,
pair: <a, b>,
pi2: snd(t),
sq_stable: SqStable(P),
so_apply: x[s],
or: P Q,
bag-separate: bag-separate(bs),
pi1: fst(t),
nil: [],
atom: Atom$n,
atom: Atom,
rec: rec(x.A[x]),
tunion: 
x:A.B[x],
b-union: A B,
filter: filter(P;l),
bag-merge: bag-merge(as;bs),
cons: [car / cdr],
MaAuto: Error :MaAuto,
p-outcome: Outcome,
RepeatFor: Error :RepeatFor,
HypSubst: Error :HypSubst,
label: ...$L... t,
nat_plus: 
,
l_contains: A B,
cmp-le: cmp-le(cmp;x;y),
inject: Inj(A;B;f),
reducible: reducible(a),
prime: prime(a),
l_exists: (xL. P[x]),
l_all: (xL.P[x]),
fun-connected: y is f*(x),
qle: r s,
qless: r < s,
q-rel: q-rel(r;x),
sq_exists: x:{A| B[x]},
exists: x:A. B[x],
i-finite: i-finite(I),
i-closed: i-closed(I),
dstype: dstype(TypeNames; d; a),
fset-member: a s,
f-subset: xs ys,
fset-closed: (s closed under fs),
MaName: MaName,
l_disjoint: l_disjoint(T;l1;l2),
decidable: Dec(P),
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),
bag-member: bag-member(T;x;bs),
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)
Lemmas :
length_wf2,
sq_stable__valueall-type,
decidable__equal_int,
length_nil,
bfalse_wf,
btrue_wf,
subtype_rel_self,
select-cons,
bnot_of_le_int,
pi2_wf,
bag-separate_wf,
pos_length2,
bag-merge_wf,
l_member_wf,
list-subtype,
unit_wf,
union-valueall-type,
df-program-type-valueall-type,
int_seg_properties,
length_cons,
non_neg_length,
pi1_wf_top,
subtype_base_sq,
int_subtype_base,
quotient_wf,
equiv_rel_wf,
trans_wf,
sym_wf,
refl_wf,
true_wf,
squash_wf,
subtype_rel_bag,
intensional-universe_wf,
nat_properties,
nat_ind_tp,
select_wf,
dataflow-program_wf,
bag_wf,
member_wf,
int_seg_wf,
df-program-type_wf,
valueall-type_wf,
uall_wf,
length_wf1,
guard_wf,
le_wf,
nat_wf,
comp_nat_ind_tp,
length_wf_nat,
top_wf,
bool_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
assert_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
le_int_wf,
bnot_wf,
lt_int_wf,
ifthenelse_wf,
eq_int_wf,
tl_wf,
hd_wf,
parallel-df-prog2_wf,
parallel-df-prog1_wf,
ge_wf,
not_wf,
false_wf,
pos_length3,
permutation_wf,
subtype_rel_wf,
assert_of_eq_int,
assert_of_bnot,
not_functionality_wrt_uiff
\mforall{}[A:\mBbbU{}']. \mforall{}[dfps:DataflowProgram(A) List].
\mforall{}[B:\{B:Type| valueall-type(B)\} ]. \mforall{}[F:k:\mBbbN{}||dfps|| {}\mrightarrow{} bag(df-program-type(dfps[k])) {}\mrightarrow{} bag(B)].
(parallel-df-program(B;F;dfps) \mmember{} DataflowProgram(A))
supposing 0 < ||dfps||
Date html generated:
2011_08_16-AM-09_44_26
Last ObjectModification:
2011_06_18-AM-08_34_47
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