{ 
[A,B,C,S:Type]. [F:S 
A (S 
C)]. [G:C B B]. [P:B 
].
[s:S]. [s2:B].
(better-feedback-dataflow(1;
k.[rec-dataflow(s;s,a.F[s;a])][k];
g,x.G[g 0;x];s2;s.P[s])
= rec-dataflow(<s, s2>;s,a.let s1,s2 = s
in let s',out = F[s1;a]
in <<s'
, if P[G[out;s2]]
then G[out;s2]
else s2
fi
>
, G[out;s2]
>)) }
{ Proof }
Definitions occuring in Statement :
better-feedback-dataflow: better-feedback-dataflow(n;ds;F;s;x.P[x]),
rec-dataflow: rec-dataflow(s0;s,m.next[s; m]),
dataflow: dataflow(A;B),
select: l[i],
ifthenelse: if b then t else f fi ,
bool: ,
uall: [x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
apply: f a,
lambda: x.A[x],
function: x:A B[x],
spread: spread def,
pair: <a, b>,
product: x:A B[x],
cons: [car / cdr],
nil: [],
natural_number: $n,
universe: Type,
equal: s = t
Definitions :
suptype: suptype(S; T),
sqequal: s ~ t,
true: True,
data-stream: data-stream(P;L),
fpf-cap: f(x)?z,
sq_stable: SqStable(P),
proper-iseg: L1 < L2,
iseg: l1 
l2,
gt: i > j,
map: map(f;as),
label: ...$L... t,
top: Top,
grp_car: |g|,
filter: filter(P;l),
intensional-universe: IType,
corec: corec(T.F[T]),
tl: tl(l),
hd: hd(l),
length: ||as||,
so_lambda: 
x y.t[x; y],
let: let,
sq_type: SQType(T),
list: type List,
nat_plus: 
,
l_contains: A 
B,
cmp-le: cmp-le(cmp;x;y),
inject: Inj(A;B;f),
reducible: reducible(a),
prime: prime(a),
squash: 
T,
l_exists: (
x
L. P[x]),
l_all: (xL.P[x]),
infix_ap: x f y,
fun-connected: y is f*(x),
limited-type: LimitedType,
qle: r s,
qless: r < s,
q-rel: q-rel(r;x),
sq_exists: x:{A| B[x]},
exists: x:A. B[x],
atom: Atom$n,
i-finite: i-finite(I),
i-closed: i-closed(I),
dstype: dstype(TypeNames; d; a),
fset-member: a s,
f-subset: xs ys,
fset: FSet{T},
fset-closed: (s closed under fs),
Id: Id,
IdLnk: IdLnk,
Knd: Knd,
MaName: MaName,
l_disjoint: l_disjoint(T;l1;l2),
decidable: Dec(P),
union: left + right,
or: P 
Q,
guard: {T},
l_member: (x l),
assert: 
b,
lelt: i j < k,
int_seg: {i..j
},
p-outcome: Outcome,
prop: 
,
void: Void,
implies: P 
Q,
false: False,
set: {x:A| B[x]} ,
real: 
,
rationals: 
,
subtype: S T,
nat: ,
int: 
,
so_lambda: x.t[x],
fpf: a:A fp-> B[a],
strong-subtype: strong-subtype(A;B),
le: A B,
ge: i 
j ,
not: 
A,
less_than: a < b,
uimplies: b supposing a,
and: P 
Q,
uiff: uiff(P;Q),
subtype_rel: A r B,
all: x:A. B[x],
axiom: Ax,
ifthenelse: if b then t else f fi ,
spread: spread def,
pair: <a, b>,
so_apply: x[s],
nil: [],
apply: f a,
so_apply: x[s1;s2],
rec-dataflow: rec-dataflow(s0;s,m.next[s; m]),
cons: [car / cdr],
select: l[i],
natural_number: $n,
better-feedback-dataflow: better-feedback-dataflow(n;ds;F;s;x.P[x]),
dataflow: dataflow(A;B),
bool: ,
equal: s = t,
universe: Type,
product: x:A B[x],
uall: [x:A]. B[x],
isect: 
x:A. B[x],
member: t T,
function: x:A B[x],
Auto: Error :Auto,
CollapseTHEN: Error :CollapseTHEN,
CollapseTHENA: Error :CollapseTHENA,
lambda: x.A[x],
Try: Error :Try
Lemmas :
member_wf,
dataflow_wf,
select_wf,
int_seg_properties,
le_wf,
false_wf,
not_wf,
int_subtype_base,
subtype_base_sq,
int_seg_wf,
decidable__equal_int,
nat_wf,
better-feedback-dataflow_wf,
dataflow-extensionality,
bool_wf,
rec-dataflow_wf,
intensional-universe_wf,
subtype_rel_wf,
length_wf_nat,
length_nil,
top_wf,
ge_wf,
subtype_rel_self,
ifthenelse_wf,
true_wf,
squash_wf,
data-stream_wf,
feedback-1-rec-dataflow
\mforall{}[A,B,C,S:Type]. \mforall{}[F:S {}\mrightarrow{} A {}\mrightarrow{} (S \mtimes{} C)]. \mforall{}[G:C {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[P:B {}\mrightarrow{} \mBbbB{}]. \mforall{}[s:S]. \mforall{}[s2:B].
(better-feedback-dataflow(1;\mlambda{}k.[rec-dataflow(s;s,a.F[s;a])][k];\mlambda{}g,x.G[g 0;x];s2;s.P[s])
= rec-dataflow(<s, s2>s,a.let s1,s2 = s
in let s',out = F[s1;a]
in <<s', if P[G[out;s2]] then G[out;s2] else s2 fi >, G[out;s2]>))
Date html generated:
2011_08_10-AM-08_20_41
Last ObjectModification:
2011_06_18-AM-08_32_06
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