{ 
[A:{A:
'| 
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-case2(B;F;dfps))
supposing f:k:||dfps|| bag(df-program-type(dfps[k]))
((
k:||dfps||. ((f k) = {})) 
((F f) = {})) }
{ Proof }
Definitions occuring in Statement :
parallel-df-program-case2: parallel-df-program-case2(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],
all: x:A. B[x],
exists: x:A. B[x],
squash: T,
implies: P Q,
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 :
true: True,
divides: b | a,
assoced: a ~ b,
set_leq: a 
b,
set_lt: a <p b,
grp_lt: a < b,
cand: A c
B,
l_contains: A 
B,
cmp-le: cmp-le(cmp;x;y),
reducible: reducible(a),
prime: prime(a),
l_exists: (x
L. 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]},
i-finite: i-finite(I),
i-closed: i-closed(I),
p-outcome: Outcome,
fset-member: a s,
f-subset: xs ys,
fset-closed: (s closed under fs),
l_disjoint: l_disjoint(T;l1;l2),
decidable: Dec(P),
uni_sat: a = !x:T. Q[x],
inv_funs: InvFuns(A;B;f;g),
inject: Inj(A;B;f),
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),
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,
sq_stable: SqStable(P),
strict-bag-function: strict-bag-function(G;L;B;S),
l_all: (xL.P[x]),
decide: case b of inl(x) => s[x] | inr(y) => t[y],
bnot: 
b,
isl: isl(x),
bl-exists: (xL.P[x])_b,
filter: filter(P;l),
list_ind: list_ind def,
permutation: permutation(T;L1;L2),
select-tuple: x.n,
sqequal: s ~ t,
so_apply: x[s],
or: P 
Q,
guard: {T},
l_member: (x l),
assert: 
b,
nil: [],
label: ...$L... t,
grp_car: |g|,
corec: corec(T.F[T]),
listp: A List
,
combination: Combination(n;T),
rev_implies: P 
Q,
iff: P Q,
atom: Atom,
rec: rec(x.A[x]),
atom: Atom$n,
Id: Id,
qeq: qeq(r;s),
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
tunion: 
x:A.B[x],
int_nzero: 
,
b-union: A B,
upto: upto(n),
quotient: x,y:A//B[x; y],
has-valueall: has-valueall(a),
callbyvalueall: callbyvalueall,
nat: ,
has-value: has-value(a),
callbyvalue: callbyvalue,
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,
parallel-df-prog: parallel-df-prog,
tuple: tuple(n;i.F[i]),
parallel-df-halt: parallel-df-halt(G;L;B;halt),
bool: 
,
unit: Unit,
top: Top,
union: left + right,
tuple-type: tuple-type(L),
and: P Q,
lelt: i j < k,
less_than: a < b,
void: Void,
false: False,
not: A,
le: A B,
empty-bag: {},
apply: f a,
limited-type: LimitedType,
real: 
,
rationals: 
,
subtype: S T,
int: ,
natural_number: $n,
axiom: Ax,
parallel-df-program-case2: parallel-df-program-case2(B;F;dfps),
df-program-meaning: df-program-meaning(dfp),
map: map(f;as),
lambda: x.A[x],
length: ||as||,
better-parallel-dataflow: better-parallel-dataflow,
member: t T,
prop: 
,
squash: T,
list: type List,
valueall-type: valueall-type(T),
int_seg: {i..j},
uall: [x:A]. B[x],
so_lambda: 
x.t[x],
uimplies: b supposing a,
isect: 
x:A. B[x],
dataflow: dataflow(A;B),
all: x:A. B[x],
implies: P Q,
function: x:A B[x],
exists: x:A. B[x],
product: x:A 
B[x],
equal: s = t,
bag: bag(T),
df-program-type: df-program-type(dfp),
select: l[i],
dataflow-program: DataflowProgram(A),
set: {x:A| B[x]} ,
universe: Type,
fpf-cap: f(x)?z,
sq_type: SQType(T),
intensional-universe: IType,
nat_plus: ,
dstype: dstype(TypeNames; d; a),
fset: FSet{T},
IdLnk: IdLnk,
Knd: Knd,
MaName: MaName,
rev_uimplies: rev_uimplies(P;Q),
tl: tl(l),
hd: hd(l),
eq_bool: p =b q,
lt_int: i <z j,
le_int: i z j,
eq_int: (i =
j),
eq_atom: x =a y,
null: null(as),
set_blt: a < b,
grp_blt: a < b,
dcdr-to-bool: [d],
bl-all: (xL.P[x])_b,
b-exists: (i<n.P[i])_b,
eq_type: eq_type(T;T'),
eq_atom: eq_atom$n(x;y),
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,
MaAuto: Error :MaAuto,
cons: [car / cdr],
CollapseTHEN: Error :CollapseTHEN,
Auto: Error :Auto,
D: Error :D,
CollapseTHENA: Error :CollapseTHENA,
THENM: Error :THENM,
AssertBY: Error :AssertBY,
fpf-dom: x dom(f),
BHyp: Error :BHyp,
tactic: Error :tactic,
n-tuple: n-tuple(n)
Lemmas :
tuple_wf,
select-tuple-tuple,
cons_member,
nat_properties,
subtype_rel_bag,
sq_stable__all,
assert-bl-exists,
decidable__l_all-better-extract,
decidable__assert,
true_wf,
assert_of_bnot,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
l_all_wf2,
sq_stable__and,
decidable__lt,
sq_stable__equal,
intensional-universe_wf,
intensional-universe_wf2,
select_wf,
dataflow-program_wf,
empty-bag_wf,
df-program-type_wf,
bag_wf,
length_wf1,
int_seg_wf,
valueall-type_wf,
squash_wf,
uall_wf,
dataflow_wf,
parallel-df-prog-meaning,
real-has-value,
int_inc_real,
rational-has-value,
upto_wf,
rationals_wf,
member_wf,
int_nzero_wf,
b-union_wf,
bool_wf,
ifthenelse_wf,
tunion_wf,
subtype_rel_wf,
int-rational,
btrue_wf,
bfalse_wf,
Id-has-valueall,
Id_wf,
list-valueall-type,
le_wf,
set-valueall-type,
int-valueall-type,
iff_wf,
rev_implies_wf,
better-parallel-dataflow_wf,
length_wf_nat,
top_wf,
map_wf,
df-program-meaning_wf,
map_length,
non_neg_length,
nat_wf,
ge_wf,
not_wf,
false_wf,
int_seg_properties,
select-map,
tuple-type_wf,
select-tuple_wf,
permutation_wf,
length-map,
unit_wf,
bl-exists_wf,
l_member_wf,
bnot_wf,
isl_wf,
parallel-df-halt_wf,
assert_wf,
l_all_wf,
strict-bag-function_wf,
sq_stable_from_decidable
\mforall{}[A:\{A:\mBbbU{}'| \mdownarrow{}A\} ]. \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-case2(B;F;dfps))
supposing \mforall{}f:k:\mBbbN{}||dfps|| {}\mrightarrow{} bag(df-program-type(dfps[k]))
((\mexists{}k:\mBbbN{}||dfps||. ((f k) = \{\})) {}\mRightarrow{} ((F f) = \{\}))
Date html generated:
2011_08_16-AM-09_39_55
Last ObjectModification:
2011_06_10-PM-03_49_34
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