Nuprl Lemma : Accum-class-progress
[Info,B,A:Type].
R:B 
B 
. P:A B . f:A B B. init:Id bag(B). X:EClass(A). es:EO+(Info). e1,e2:E. v1,v2:B.
((a:A. s:B. Dec(P[a;s]))
Trans(B;x,y.R[x;y])
(s1,s2:B. SqStable(R[s1;s2]))
(a:A. e:E. s:B.
((e1 <loc e)
e 
loc e2
a 
X(e)
s Prior(Accum-class(f;init;X))?init(e)
((P[a;s] R[s;f a s]) 
((
P[a;s]) (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)
(((
e:E. a:A. s:B. ((e1 <loc e) e loc e2 s Prior(Accum-class(f;init;X))?init(e) a X(e) P[a;s]))
R[v1;v2])
((e:E. s:B.
((e1 <loc e)
e loc e2
s Prior(Accum-class(f;init;X))?init(e)
(a:A. ((a X(e)) 
(P[a;s])))))
(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),
decidable: Dec(P),
uall: [x:A]. B[x],
prop: ,
so_apply: x[s1;s2],
all: x:A. B[x],
exists: x:A. B[x],
not: A,
implies: P Q,
or: P Q,
and: P Q,
apply: f a,
function: x:A B[x],
universe: Type,
equal: s = t,
single-valued-bag: single-valued-bag(b;T),
bag: bag(T)
Definitions :
so_lambda: 
x y.t[x; y],
iff: P 
Q,
rev_implies: P Q,
so_lambda: x.t[x],
top: Top,
true: True,
squash: 
T,
false: False,
le: A B,
ge: i 
j ,
nat: 
,
member: t T,
or: P Q,
exists: x:A. B[x],
not: A,
and: P Q,
sq_stable: SqStable(P),
so_apply: x[s1;s2],
implies: P Q,
prop: ,
all: x:A. B[x],
uall: [x:A]. B[x],
cand: A c B,
guard: {T},
es-le: e loc e' ,
Accum-class: Accum-class(f;init;X),
so_apply: x[s],
alle-at: e@i.P[e],
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: 
b,
decidable: Dec(P),
es-p-local-pred: es-p-local-pred(es;P),
uiff: uiff(P;Q),
strongwellfounded: SWellFounded(R[x; y]),
es-locl: (e <loc e'),
trans: Trans(T;x,y.E[x; y]),
bag-member: x bs,
classrel: v X(e),
rev_uimplies: rev_uimplies(P;Q),
single-valued-classrel: single-valued-classrel(es;X;T),
sq_type: SQType(T),
existse-between3: e(e1,e2].P[e],
subtype: S 
T,
Memory-class: Memory-class(f;init;X)
Lemmas :
bag_wf,
Id_wf,
event-ordering+_wf,
eclass_wf,
event-ordering+_inc,
es-E_wf,
decidable_wf,
trans_wf,
sq_stable_wf,
single-valued-classrel_wf,
es-loc_wf,
single-valued-bag_wf,
Accum-class_wf,
es-locl-trichotomy,
primed-class-opt-classrel,
squash_wf,
sq_stable__equal,
sq_stable__all,
equal_wf,
not_wf,
or_wf,
all_wf,
classrel_wf,
es-le_wf,
es-locl_wf,
and_wf,
exists_wf,
sq_stable__and,
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,
Prior-Accum-class-single-val0,
es-first_wf,
assert_wf,
decidable__existse-between3,
or_false_l,
and_false_l,
or_functionality_wrt_iff,
Memory-classrel,
and_functionality_wrt_iff,
exists_functionality_wrt_iff,
decidable_functionality,
es-loc-pred,
decidable__exists_iterated_classrel,
false_wf,
not_assert_elim,
bool_subtype_base,
bool_wf,
subtype_base_sq,
es-pred_wf,
iterated_classrel_wf,
bag-member_wf,
Memory-class_wf,
decidable__exists-single-valued-classrel,
iterated_classrel-single-val,
true_wf,
Accum-class-single-val0
\mforall{}[Info,B,A:Type].
\mforall{}R:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}. \mforall{}P:A {}\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.
((\mforall{}a:A. \mforall{}s:B. Dec(P[a;s]))
{}\mRightarrow{} Trans(B;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}s1,s2:B. SqStable(R[s1;s2]))
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E. \mforall{}s:B.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} s \mmember{} Prior(Accum-class(f;init;X))?init(e)
{}\mRightarrow{} ((P[a;s] {}\mRightarrow{} R[s;f a s]) \mwedge{} ((\mneg{}P[a;s]) {}\mRightarrow{} (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{} (((\mexists{}e:E
\mexists{}a:A
\mexists{}s:B
((e1 <loc e)
\mwedge{} e \mleq{}loc e2
\mwedge{} s \mmember{} Prior(Accum-class(f;init;X))?init(e)
\mwedge{} a \mmember{} X(e)
\mwedge{} P[a;s]))
{}\mRightarrow{} R[v1;v2])
\mwedge{} ((\mforall{}e:E. \mforall{}s:B.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} s \mmember{} Prior(Accum-class(f;init;X))?init(e)
{}\mRightarrow{} (\mforall{}a:A. ((\mneg{}a \mmember{} X(e)) \mvee{} (\mneg{}P[a;s])))))
{}\mRightarrow{} (v1 = v2))))
Date html generated:
2012_02_20-PM-03_02_06
Last ObjectModification:
2012_02_07-AM-01_52_18
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