Nuprl Lemma : feedback-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 C].
[buf:C]. [P:C 
].
(feedback-dataflow(ds;
F;buf;x.P[x])
= feedback-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 ));buf;x.P[x]))
Proof not projected
Definitions occuring in Statement :
feedback-dataflow: feedback-dataflow,
better-parallel-dataflow: better-parallel-dataflow,
dataflow: dataflow(A;B),
eq_int: (i = j),
ifthenelse: if b then t else f fi ,
bool: ,
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},
bag: bag(T),
so_apply: x[s],
ifthenelse: if b then t else f fi ,
eq_int: (i = j),
pi2: snd(t),
member: t 
T,
le: A 
B,
not: 
A,
implies: P 
Q,
false: False,
so_lambda: 
x.t[x],
squash: 
T,
true: True,
lelt: i j < k,
and: P 
Q,
all: x:A. B[x],
btrue: tt,
subtype: S 
T,
suptype: suptype(S; T),
bfalse: ff,
guard: {T},
top: Top,
let: let,
so_lambda: x y.t[x; y],
parallel-dataflow: parallel-dataflow,
pi1: fst(t),
bool: ,
prop: 
,
unit: Unit,
uimplies: b supposing a,
uiff: uiff(P;Q),
so_apply: x[s1;s2],
sq_type: SQType(T),
or: P 
Q,
it: 
Lemmas :
feedback-dataflow-equal,
le_wf,
bag_wf,
equal_wf,
squash_wf,
true_wf,
dataflow_wf,
ifthenelse_wf,
eq_int_wf,
lelt_wf,
int_seg_wf,
bool_wf,
uiff_transitivity,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
better-parallel-dataflow_wf,
bag-merge_wf,
bnot_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
not_functionality_wrt_uiff,
subtype_rel_self,
pi1_wf_top,
bag-separate_wf,
subtype_rel_simple_product,
top_wf,
permutation_wf,
nat_wf,
eq_int_eq_true,
and_wf,
dataflow-extensionality,
parallel-dataflow_wf,
rec-dataflow_wf,
dataflow-ap_wf,
subtype_base_sq,
int_subtype_base,
pi2_wf,
data-stream-cons,
dataflow-out_wf,
bool_cases,
bool_subtype_base,
parallel-bag-dataflow-recode,
let_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 {}\mrightarrow{} C]. \mforall{}[buf:C]. \mforall{}[P:C {}\mrightarrow{} \mBbbB{}].
(feedback-dataflow(ds;
F;buf;x.P[x])
= feedback-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 ));buf;x.P[x]))
Date html generated:
2012_01_23-AM-11_58_13
Last ObjectModification:
2011_12_28-PM-03_08_17
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