Nuprl Lemma : parallel-bag-dataflow-recode
[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].
(parallel-dataflow(
ds;
F)
= parallel-dataflow(
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 not projected
Definitions occuring in Statement :
better-parallel-dataflow: better-parallel-dataflow,
parallel-dataflow: 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,
bag-separate: bag-separate(bs),
bag-merge: bag-merge(as;bs),
bag: bag(T)
Definitions :
uall: [x:A]. B[x],
nat: ,
int_seg: {i..j},
so_apply: x[s],
parallel-dataflow: parallel-dataflow,
ifthenelse: if b then t else f fi ,
pi1: fst(t),
pi2: snd(t),
member: t 
T,
prop: 
,
all: x:A. B[x],
implies: P 
Q,
lelt: i 
j < k,
and: P 
Q,
le: A B,
not: 
A,
false: False,
so_lambda: 
x y.t[x; y],
top: Top,
subtype: S 
T,
btrue: tt,
bfalse: ff,
guard: {T},
squash: 
T,
true: True,
so_lambda: x.t[x],
suptype: suptype(S; T),
uimplies: b supposing a,
so_apply: x[s1;s2],
bool: 
,
unit: Unit,
uiff: uiff(P;Q),
sq_type: SQType(T),
or: P 
Q,
it: 
Lemmas :
and_wf,
rec-dataflow-equal,
int_seg_wf,
dataflow_wf,
bag_wf,
ifthenelse_wf,
eq_int_wf,
lelt_wf,
pi1_wf_top,
dataflow-ap_wf,
dataflow-out_wf,
bool_wf,
uiff_transitivity,
equal_wf,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
bnot_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
not_functionality_wrt_uiff,
squash_wf,
true_wf,
better-parallel-dataflow_wf,
le_wf,
bag-merge_wf,
nat_wf,
eq_int_eq_true,
bool_subtype_base,
subtype_base_sq,
bool_cases
\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].
(parallel-dataflow(
ds;
F)
= parallel-dataflow(
\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:
2012_01_23-AM-11_57_51
Last ObjectModification:
2011_12_17-PM-04_32_37
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