Nuprl Lemma : parallel-dataflow-recode
[A,C:Type]. [n:
]. [B:n + 2 
Type]. [ds:k:n + 2 dataflow(A;B[k])]. [F:k:n + 2 B[k] C].
(parallel-dataflow(
ds;
F)
= parallel-dataflow(
k.if (k =
0) then better-parallel-dataflow(2;ds;s.<s 0, s 1>) else ds (k + 1) fi ;
g.(F (k.if (k = 0) then fst((g 0)) if (k = 1) then snd((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],
pair: <a, b>,
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t
Definitions :
suptype: suptype(S; T),
so_lambda: 
x.t[x],
true: True,
squash: 
T,
guard: {T},
bfalse: ff,
btrue: tt,
subtype: S 
T,
top: Top,
so_lambda: x y.t[x; y],
false: False,
not: 
A,
le: A 
B,
and: P 
Q,
lelt: i j < k,
implies: P 
Q,
all: x:A. B[x],
prop: 
,
member: t 
T,
pi2: snd(t),
pi1: fst(t),
ifthenelse: if b then t else f fi ,
parallel-dataflow: parallel-dataflow,
so_apply: x[s],
int_seg: {i..j},
nat: ,
uall: [x:A]. B[x],
uiff: uiff(P;Q),
unit: Unit,
bool: 
,
so_apply: x[s1;s2],
uimplies: b supposing a,
sq_type: SQType(T),
or: P 
Q,
it: 
Lemmas :
true_wf,
squash_wf,
and_wf,
eq_int_eq_true,
nat_wf,
le_wf,
better-parallel-dataflow_wf,
top_wf,
subtype_rel_simple_product,
not_functionality_wrt_uiff,
assert_of_bnot,
eqff_to_assert,
not_wf,
bnot_wf,
assert_of_eq_int,
eqtt_to_assert,
assert_wf,
equal_wf,
uiff_transitivity,
bool_wf,
dataflow-out_wf,
dataflow-ap_wf,
pi1_wf_top,
lelt_wf,
eq_int_wf,
ifthenelse_wf,
dataflow_wf,
int_seg_wf,
rec-dataflow-equal,
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;B[k])].
\mforall{}[F:k:\mBbbN{}n + 2 {}\mrightarrow{} B[k] {}\mrightarrow{} C].
(parallel-dataflow(
ds;
F)
= parallel-dataflow(
\mlambda{}k.if (k =\msubz{} 0) then better-parallel-dataflow(2;ds;\mlambda{}s.<s 0, s 1>) else ds (k + 1) fi ;
\mlambda{}g.(F (\mlambda{}k.if (k =\msubz{} 0) then fst((g 0)) if (k =\msubz{} 1) then snd((g 0)) else g (k - 1) fi ))))
Date html generated:
2012_01_23-AM-11_57_42
Last ObjectModification:
2011_12_13-AM-10_12_55
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