{ 
[Info,A:Type]. [a:A]. [X:EClass(A)]. [S:Type]. [init:S].
[f:S 
A S]. [test:S A 
]. [nxt:S 
A S]. [B:Type].
[g:S A (B + Top)].
(g[Threshold(init;f;test;nxt;X)]
= 
p.(snd(p))[es-interface-accum(p,x.
let s',v = p
in if test s' x
then <nxt <s', x>, g <s', x>>
else <f s' x, ff>
fi ;<init
, if test init a
then g <init, a>
else ff
fi
>;X)]) }
{ Proof }
Definitions occuring in Statement :
es-threshold: Threshold(init;f;test;nxt;X),
es-interface-accum: es-interface-accum(f;x;X),
es-filter-image: f[X],
eclass: EClass(A[eo; e]),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bool: ,
uall: [x:A]. B[x],
top: Top,
pi2: snd(t),
apply: f a,
lambda: x.A[x],
function: x:A B[x],
spread: spread def,
pair: <a, b>,
product: x:A B[x],
union: left + right,
universe: Type,
equal: s = t
Definitions :
uall: [x:A]. B[x],
eclass: EClass(A[eo; e]),
top: Top,
es-filter-image: f[X],
pi2: snd(t),
es-interface-accum: es-interface-accum(f;x;X),
ifthenelse: if b then t else f fi ,
bfalse: ff,
spreadn: spread3,
member: t 
T,
all: x:A. B[x],
implies: P 
Q,
btrue: tt,
prop: 
,
and: P 
Q,
pi1: fst(t),
so_lambda: 
x.t[x],
p-compose: f o g,
can-apply: can-apply(f;x),
do-apply: do-apply(f;x),
isl: isl(x),
outl: outl(x),
subtype: S 
T,
suptype: suptype(S; T),
so_lambda: x y.t[x; y],
list_accum: list_accum(x,a.f[x; a];y;l),
ycomb: Y,
bool: ,
unit: Unit,
iff: P 
Q,
so_apply: x[s],
es-E-interface: E(X),
so_apply: x[s1;s2],
it: 
Lemmas :
es-threshold-as-accum,
bool_wf,
iff_weakening_uiff,
assert_wf,
eqtt_to_assert,
not_wf,
uiff_transitivity,
bnot_wf,
eqff_to_assert,
assert_of_bnot,
top_wf,
es-filter-image_wf,
es-interface-accum_wf,
ifthenelse_wf,
pi2_wf,
in-eclass_wf,
event-ordering+_wf,
es-E_wf,
event-ordering+_inc,
es-interface-predecessors_wf,
es-E-interface_wf,
Id_wf,
es-loc_wf,
eclass_wf,
es-interface-val_wf2,
pi1_wf_top
\mforall{}[Info,A:Type]. \mforall{}[a:A]. \mforall{}[X:EClass(A)]. \mforall{}[S:Type]. \mforall{}[init:S]. \mforall{}[f:S {}\mrightarrow{} A {}\mrightarrow{} S]. \mforall{}[test:S {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}].
\mforall{}[nxt:S \mtimes{} A {}\mrightarrow{} S]. \mforall{}[B:Type]. \mforall{}[g:S \mtimes{} A {}\mrightarrow{} (B + Top)].
(g[Threshold(init;f;test;nxt;X)]
= \mlambda{}p.(snd(p))[es-interface-accum(\mlambda{}p,x.
let s',v = p
in if test s' x
then <nxt <s', x>, g <s', x>>
else <f s' x, ff>
fi ;<init, if test init a then g <init, a> else ff fi >X)])
Date html generated:
2011_08_16-PM-06_08_33
Last ObjectModification:
2010_11_10-AM-05_41_52
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