Nuprl Lemma : Accum-class-trans
[Info,B,A:Type].
R:B 
B 
. f:A B B. init:Id bag(B). X:EClass(A). es:EO+(Info). e1,e2:E. v1,v2:B.
(Trans(B;x,y.R[x;y])
(s1,s2:B. SqStable(R[s1;s2]))
(a:A. e:E.
((e1 <loc e) e 
loc e2 a 
X(e) (s:B. (s Prior(Accum-class(f;init;X))?init(e) R[s;f a s]))))
single-valued-classrel(es;X;A)
single-valued-bag(init loc(e1);B)
v1 Accum-class(f;init;X)(e1)
v2 Accum-class(f;init;X)(e2)
(e1 <loc e2)
R[v1;v2])
Proof not projected
Definitions occuring in Statement :
Accum-class: Accum-class(f;init;X),
primed-class-opt: Prior(X)?b,
single-valued-classrel: single-valued-classrel(es;X;T),
classrel: v X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-le: e loc e' ,
es-locl: (e <loc e'),
es-loc: loc(e),
es-E: E,
Id: Id,
trans: Trans(T;x,y.E[x; y]),
sq_stable: SqStable(P),
uall: [x:A]. B[x],
prop: ,
so_apply: x[s1;s2],
all: x:A. B[x],
implies: P Q,
apply: f a,
function: x:A B[x],
universe: Type,
single-valued-bag: single-valued-bag(b;T),
bag: bag(T)
Definitions :
so_lambda: 
x y.t[x; y],
so_lambda: x.t[x],
iff: P 
Q,
rev_implies: P Q,
and: P 
Q,
top: Top,
true: True,
squash: 
T,
false: False,
not: 
A,
le: A B,
ge: i 
j ,
nat: 
,
member: t T,
so_apply: x[s1;s2],
implies: P Q,
prop: ,
all: x:A. B[x],
uall: [x:A]. B[x],
cand: A c B,
es-p-local-pred: es-p-local-pred(es;P),
guard: {T},
or: P 
Q,
exists: 
x:A. B[x],
es-le: e loc e' ,
so_apply: x[s],
uiff: uiff(P;Q),
uimplies: b supposing a,
strongwellfounded: SWellFounded(R[x; y]),
Accum-class: Accum-class(f;init;X),
es-locl: (e <loc e'),
sq_stable: SqStable(P),
trans: Trans(T;x,y.E[x; y]),
rev_uimplies: rev_uimplies(P;Q),
subtype: S 
T
Lemmas :
bag_wf,
Id_wf,
event-ordering+_wf,
eclass_wf,
event-ordering+_inc,
es-E_wf,
equal_wf,
trans_wf,
sq_stable_wf,
es-le_wf,
all_wf,
single-valued-classrel_wf,
es-loc_wf,
single-valued-bag_wf,
Accum-class_wf,
classrel_wf,
es-locl_wf,
es-locl-trichotomy,
primed-class-opt-classrel,
lifting-2_wf,
rec-combined-class-opt-1_wf,
primed-class-opt_wf,
simple-comb-2-classrel,
rec-combined-class-opt-1-classrel,
es-causl_wf,
le_wf,
nat_wf,
less_than_wf,
ge_wf,
nat_properties,
es-causl-swellfnd,
es-le_weakening,
es-locl_transitivity1,
bag-member_wf,
not_wf,
exists_wf,
squash_wf,
es-p-local-pred_wf,
and_wf,
Accum-class-single-val0
\mforall{}[Info,B,A:Type].
\mforall{}R:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}. \mforall{}f:A {}\mrightarrow{} B {}\mrightarrow{} B. \mforall{}init:Id {}\mrightarrow{} bag(B). \mforall{}X:EClass(A). \mforall{}es:EO+(Info). \mforall{}e1,e2:E.
\mforall{}v1,v2:B.
(Trans(B;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}s1,s2:B. SqStable(R[s1;s2]))
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} (\mforall{}s:B. (s \mmember{} Prior(Accum-class(f;init;X))?init(e) {}\mRightarrow{} R[s;f a s]))))
{}\mRightarrow{} single-valued-classrel(es;X;A)
{}\mRightarrow{} single-valued-bag(init loc(e1);B)
{}\mRightarrow{} v1 \mmember{} Accum-class(f;init;X)(e1)
{}\mRightarrow{} v2 \mmember{} Accum-class(f;init;X)(e2)
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} R[v1;v2])
Date html generated:
2012_02_20-PM-03_01_32
Last ObjectModification:
2012_02_06-PM-11_48_13
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