{ 
[A,C:Type]. [n:
]. [B:n + 2 
Type].
[ds:k:n + 2 dataflow(A;bag(B[k]))]. [F:k:n + 2 bag(B[k]) C].
(better-parallel-dataflow(
n + 2;ds;
F)
= better-parallel-dataflow(
n + 1;
k.if (k =
0)
then better-parallel-dataflow(
2;ds;
s.bag-merge(s 0;s 1))
else ds (k + 1)
fi ;
g.(F
(k.if (k = 0) then fst(bag-separate(g 0))
if (k = 1) then snd(bag-separate(g 0))
else g (k - 1)
fi )))) }
{ Proof }
Definitions occuring in Statement :
better-parallel-dataflow: better-parallel-dataflow,
dataflow: dataflow(A;B),
eq_int: (i = j),
ifthenelse: if b then t else f fi ,
int_seg: {i..j
},
nat: ,
uall: [x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
apply: f a,
lambda: x.A[x],
function: x:A B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t
Definitions :
pair: <a, b>,
fpf: a:A fp-> B[a],
strong-subtype: strong-subtype(A;B),
ge: i 
j ,
less_than: a < b,
uimplies: b supposing a,
product: x:A 
B[x],
and: P 
Q,
uiff: uiff(P;Q),
subtype_rel: A 
r B,
implies: P 
Q,
false: False,
not: 
A,
le: A 
B,
limited-type: LimitedType,
prop: 
,
parallel-dataflow: parallel-dataflow,
set: {x:A| B[x]} ,
real: 
,
grp_car: |g|,
subtype: S T,
all: x:A. B[x],
int: 
,
axiom: Ax,
subtract: n - m,
pi2: snd(t),
bag-separate: Error :bag-separate,
pi1: fst(t),
bag-merge: Error :bag-merge,
eq_int: (i = j),
ifthenelse: if b then t else f fi ,
lambda: x.A[x],
natural_number: $n,
add: n + m,
better-parallel-dataflow: better-parallel-dataflow,
member: t 
T,
nat: ,
universe: Type,
isect: 
x:A. B[x],
uall: [x:A]. B[x],
so_lambda: 
x.t[x],
equal: s = t,
bag: Error :bag,
so_apply: x[s],
apply: f a,
int_seg: {i..j},
function: x:A B[x],
dataflow: dataflow(A;B),
Auto: Error :Auto,
CollapseTHEN: Error :CollapseTHEN,
RepeatFor: Error :RepeatFor,
upto: upto(n),
map: map(f;as),
top: Top,
sqequal: s ~ t,
data-stream: data-stream(P;L),
list: type List,
void: Void,
rationals: 
,
true: True,
squash: 
T,
deq: EqDecider(T),
ma-state: State(ds),
fpf-cap: f(x)?z,
minus: -n,
refl: Refl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
so_lambda: x y.t[x; y],
filter: filter(P;l),
permutation: permutation(T;L1;L2),
sq_type: SQType(T),
sq_stable: SqStable(P),
IdLnk: IdLnk,
Id: Id,
append: as @ bs,
locl: locl(a),
Knd: Knd,
lt_int: i <z j,
le_int: i z j,
bfalse: ff,
btrue: tt,
eq_atom: x =a y,
null: null(as),
set_blt: a <
b,
grp_blt: a < b,
infix_ap: x f y,
dcdr-to-bool: [d],
bl-all: (xL.P[x])_b,
bl-exists: (
xL.P[x])_b,
b-exists: (i<n.P[i])_b,
eq_type: eq_type(T;T'),
eq_atom: eq_atom$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),
eq_id: a = b,
eq_lnk: a = b,
bimplies: p q,
band: p q,
bor: p 
q,
bnot: b,
unit: Unit,
bool: 
,
or: P Q,
guard: {T},
l_member: (x l),
assert: 
b,
quotient: x,y:A//B[x; y],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
suptype: suptype(S; T),
p-outcome: Outcome,
lelt: i j < k,
MaAuto: Error :MaAuto,
union: left + right,
fpf-dom: x dom(f),
intensional-universe: IType,
inr: inr x ,
inl: inl x ,
int_eq: if a=b then c else d,
corec: corec(T.F[T]),
SplitOn: Error :SplitOn,
Try: Error :Try
Lemmas :
subtype_base_sq,
bool_subtype_base,
btrue_wf,
bfalse_wf,
unit_wf,
intensional-universe_wf,
eq_int_eq_true,
subtype_rel_self,
nat_properties,
int_seg_properties,
eq_int_wf,
ifthenelse_wf,
Error :bag-merge_wf,
subtype_rel_wf,
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,
pi1_wf_top,
permutation_wf,
quotient_wf,
equiv_rel_wf,
trans_wf,
sym_wf,
refl_wf,
true_wf,
squash_wf,
Error :bag-separate_wf,
subtype_rel_simple_product,
pi2_wf,
top_wf,
member_wf,
parallel-data-stream,
data-stream_wf,
le_wf,
false_wf,
not_wf,
dataflow-extensionality,
upto_wf,
map_wf,
better-parallel-data-stream,
uall_wf,
nat_wf,
int_seg_wf,
dataflow_wf,
Error :bag_wf,
parallel-bag-dataflow-recode,
better-parallel-dataflow_wf,
parallel-dataflow_wf
\mforall{}[A,C:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[B:\mBbbN{}n + 2 {}\mrightarrow{} Type]. \mforall{}[ds:k:\mBbbN{}n + 2 {}\mrightarrow{} dataflow(A;bag(B[k]))].
\mforall{}[F:k:\mBbbN{}n + 2 {}\mrightarrow{} bag(B[k]) {}\mrightarrow{} C].
(better-parallel-dataflow(
n + 2;ds;
F)
= better-parallel-dataflow(
n + 1;\mlambda{}k.if (k =\msubz{} 0)
then better-parallel-dataflow(
2;ds;
\mlambda{}s.bag-merge(s 0;s 1))
else ds (k + 1)
fi ;
\mlambda{}g.(F
(\mlambda{}k.if (k =\msubz{} 0) then fst(bag-separate(g 0))
if (k =\msubz{} 1) then snd(bag-separate(g 0))
else g (k - 1)
fi ))))
Date html generated:
2011_08_10-AM-08_23_35
Last ObjectModification:
2011_06_18-AM-08_32_55
Home
Index