{ 
[A:
']. [dfps:DataflowProgram(A) List].
[B:{B:Type| valueall-type(B)} ].
[F:k:
||dfps|| 
bag(df-program-type(dfps[k])) bag(B)].
better-parallel-dataflow(
||dfps||;
k.map(dfp.df-program-meaning(dfp);dfps)[k];
F)
= df-program-meaning(parallel-df-program(B;F;dfps))
supposing (F (i.{})) = {}
supposing 0 < ||dfps|| }
{ Proof }
Definitions occuring in Statement :
parallel-df-program: parallel-df-program(B;F;dfps),
df-program-meaning: df-program-meaning(dfp),
df-program-type: df-program-type(dfp),
dataflow-program: DataflowProgram(A),
better-parallel-dataflow: better-parallel-dataflow,
dataflow: dataflow(A;B),
select: l[i],
map: map(f;as),
length: ||as||,
int_seg: {i..j
},
uimplies: b supposing a,
uall: [x:A]. B[x],
less_than: a < b,
set: {x:A| B[x]} ,
apply: f a,
lambda: x.A[x],
function: x:A B[x],
list: type List,
natural_number: $n,
universe: Type,
equal: s = t,
empty-bag: {},
bag: bag(T),
valueall-type: valueall-type(T)
Definitions :
par_df1: par_df1{par_df1_compseq_tag_def:o}(dfp; F; B),
bag-merge: bag-merge(as;bs),
union: left + right,
parallel-df-prog2: parallel-df-prog2(B;G;dfp1;dfp2),
subtract: n - m,
pi2: snd(t),
bag-separate: bag-separate(bs),
pi1: fst(t),
ifthenelse: if b then t else f fi ,
cons: [car / cdr],
add: n + m,
parallel-df-prog1: parallel-df-prog1(B;G;dfp),
nil: [],
eq_int: (i =
j),
lt_int: i <z j,
tl: tl(l),
hd: hd(l),
pair: <a, b>,
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,
natural_number: $n,
real: 
,
rationals: 
,
subtype: S T,
int: 
,
limited-type: LimitedType,
all: x:A. B[x],
empty-bag: {},
apply: f a,
axiom: Ax,
parallel-df-program: parallel-df-program(B;F;dfps),
df-program-meaning: df-program-meaning(dfp),
map: map(f;as),
lambda: x.A[x],
length: ||as||,
better-parallel-dataflow: better-parallel-dataflow,
prop: 
,
member: t 
T,
so_lambda: 
x.t[x],
uimplies: b supposing a,
dataflow: dataflow(A;B),
equal: s = t,
function: x:A B[x],
bag: bag(T),
df-program-type: df-program-type(dfp),
select: l[i],
int_seg: {i..j},
set: {x:A| B[x]} ,
valueall-type: valueall-type(T),
less_than: a < b,
list: type List,
dataflow-program: DataflowProgram(A),
universe: Type,
isect: 
x:A. B[x],
uall: [x:A]. B[x],
minus: -n,
so_apply: x[s],
permutation: permutation(T;L1;L2),
quotient: x,y:A//B[x; y],
sq_type: SQType(T),
true: True,
assert: 
b,
is_list_splitting: is_list_splitting(T;L;LL;L2;f),
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
bag-member: bag-member(T;x;bs),
req: x = y,
rnonneg: rnonneg(r),
rleq: x y,
i-member: r I,
partitions: partitions(I;p),
modulus-of-ccontinuity: modulus-of-ccontinuity(omega;I;f),
fpf-sub: f g,
squash: 
T,
sq_stable: SqStable(P),
bool: 
,
MaAuto: Error :MaAuto,
tactic: Error :tactic,
corec: corec(T.F[T]),
p-outcome: Outcome,
or: P 
Q,
nat_plus: 
,
l_contains: A B,
cmp-le: cmp-le(cmp;x;y),
inject: Inj(A;B;f),
reducible: reducible(a),
prime: prime(a),
l_exists: (
xL. P[x]),
l_all: (xL.P[x]),
infix_ap: x f y,
fun-connected: y is f*(x),
qle: r s,
qless: r < s,
q-rel: q-rel(r;x),
sq_exists: x:{A| B[x]},
exists: x:A. B[x],
atom: Atom$n,
i-finite: i-finite(I),
i-closed: i-closed(I),
dstype: dstype(TypeNames; d; a),
fset-member: a s,
l_disjoint: l_disjoint(T;l1;l2),
MaName: MaName,
Knd: Knd,
IdLnk: IdLnk,
Id: Id,
fset-closed: (s closed under fs),
fset: FSet{T},
decidable: Dec(P),
f-subset: xs ys,
label: ...$L... t,
l_member: (x l),
cand: A c B,
iff: P 
Q,
uni_sat: a = !x:T. Q[x],
inv_funs: InvFuns(A;B;f;g),
eqfun_p: IsEqFun(T;eq),
refl: Refl(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
usym: UniformlySym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
connex: Connex(T;x,y.R[x; y]),
uconnex: uconnex(T; x,y.R[x; y]),
coprime: CoPrime(a,b),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
comm: Comm(T;op),
inverse: Inverse(T;op;id;inv),
bilinear: BiLinear(T;pl;tm),
bilinear_p: IsBilinear(A;B;C;+a;+b;+c;f),
action_p: IsAction(A;x;e;S;f),
dist_1op_2op_lr: Dist1op2opLR(A;1op;2op),
fun_thru_1op: fun_thru_1op(A;B;opa;opb;f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
cancel: Cancel(T;S;op),
monot: monot(T;x,y.R[x; y];f),
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
monoid_hom_p: IsMonHom{M1,M2}(f),
grp_leq: a b,
integ_dom_p: IsIntegDom(r),
prime_ideal_p: IsPrimeIdeal(R;P),
no_repeats: no_repeats(T;l),
value-type: value-type(T),
so_apply: x[s1;s2],
le_int: i z j,
bfalse: ff,
btrue: tt,
eq_atom: x =a y,
null: null(as),
set_blt: a <
b,
grp_blt: a < b,
dcdr-to-bool: [d],
bl-all: (xL.P[x])_b,
bl-exists: (xL.P[x])_b,
b-exists: (i<n.P[i])_b,
eq_type: eq_type(T;T'),
eq_atom: eq_atom$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,
bnot: b,
unit: Unit,
filter: filter(P;l),
sqequal: s ~ t,
listp: A List,
combination: Combination(n;T),
fpf-cap: f(x)?z,
suptype: suptype(S; T),
parameter: parm{i},
atom: Atom,
rec: rec(x.A[x]),
tunion: 
x:A.B[x],
b-union: A B,
CollapseTHEN: Error :CollapseTHEN,
Complete: Error :Complete,
Try: Error :Try,
RepeatFor: Error :RepeatFor,
Auto: Error :Auto,
CollapseTHENA: Error :CollapseTHENA,
bag-mapfilter: bag-mapfilter(f;P;bs),
bag-filter: [xb|p[x]],
bag-map: bag-map(f;bs),
it: 
,
intensional-universe: IType,
primrec: primrec(n;b;c),
proper-iseg: L1 < L2,
iseg: l1 l2,
gt: i > j
Lemmas :
list-subtype,
l_member_wf,
corec_wf,
map_length,
intensional-universe_wf,
unit_wf,
subtype_rel_self,
bag-separate_wf,
pi2_wf,
pi1_wf_top,
parallel-df-prog2_wf,
bag-merge_wf,
union-valueall-type,
df-program-type-valueall-type,
better-parallel-bag-dataflow-recode,
sq_stable__subtype_rel,
map_wf,
select-cons,
select-map,
bnot_of_le_int,
lt_int_wf,
le_int_wf,
assert_of_le_int,
bnot_of_lt_int,
assert_functionality_wrt_uiff,
assert_of_lt_int,
set_subtype_base,
int_seg_properties,
eq_int_wf,
ifthenelse_wf,
parallel-df-prog2-meaning,
bool_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
assert_wf,
eqff_to_assert,
assert_of_bnot,
not_functionality_wrt_uiff,
bnot_wf,
bfalse_wf,
btrue_wf,
sq_stable__and,
sq_stable__equal,
pos_length2,
non_neg_length,
length_cons,
length_nil,
decidable__equal_int,
false_wf,
not_wf,
true_wf,
squash_wf,
better-parallel-dataflow_wf,
df-program-meaning_wf,
sq_stable__valueall-type,
subtype_base_sq,
int_subtype_base,
parallel-df-prog1-meaning,
permutation_wf,
subtype_rel_wf,
ge_wf,
nat_properties,
nat_ind_tp,
bag_wf,
dataflow_wf,
uall_wf,
dataflow-program_wf,
valueall-type_wf,
int_seg_wf,
df-program-type_wf,
select_wf,
length_wf1,
empty-bag_wf,
guard_wf,
le_wf,
nat_wf,
comp_nat_ind_tp,
length_wf_nat,
top_wf,
member_wf
\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)].
better-parallel-dataflow(
||dfps||;\mlambda{}k.map(\mlambda{}dfp.df-program-meaning(dfp);dfps)[k];
F)
= df-program-meaning(parallel-df-program(B;F;dfps))
supposing (F (\mlambda{}i.\{\})) = \{\}
supposing 0 < ||dfps||
Date html generated:
2011_08_16-AM-09_45_03
Last ObjectModification:
2011_06_18-AM-08_34_51
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