Nuprl Lemma : iterated_classrel_trans3
[Info,A,S:Type]. [init:Id 
bag(S)]. [f:A S S]. [X:EClass(A)].
es:EO+(Info)
(single-valued-classrel(es;X;A)
([R:S S 
]
((x,y:S. SqStable(R[x;y]))
Trans(S;x,y.R[x;y])
(e1,e2:E.
((a:A. e:E.
((e1 <loc e)
e 
loc e2
a 
X(e)
(s:S. (iterated_classrel(es;S;A;f;init;X;pred(e);s) R[s;f a s]))))
single-valued-bag(init loc(e1);S)
(e1 <loc e2)
(v1,v2:S.
(iterated_classrel(es;S;A;f;init;X;e1;v1)
iterated_classrel(es;S;A;f;init;X;e2;v2)
(((
e:E. ((e1 <loc e) 
e loc e2 (a:A. a X(e)))) R[v1;v2])
((e:E. ((e1 <loc e) e loc e2 (a:A. (
a X(e))))) (v1 = v2))))))))))
Proof not projected
Definitions occuring in Statement :
iterated_classrel: iterated_classrel(es;S;A;f;init;X;e;v),
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-pred: pred(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],
exists: x:A. B[x],
not: A,
implies: 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],
bfalse: ff,
assert: 
b,
so_lambda: x.t[x],
ifthenelse: if b then t else f fi ,
true: True,
squash: 
T,
false: False,
le: A B,
ge: i 
j ,
nat: 
,
member: t T,
not: A,
exists: x:A. B[x],
and: P Q,
so_apply: x[s1;s2],
sq_stable: SqStable(P),
prop: ,
implies: P Q,
all: x:A. B[x],
uall: [x:A]. B[x],
iff: P 
Q,
rev_implies: P Q,
guard: {T},
cand: A c B,
or: P 
Q,
es-le: e loc e' ,
uiff: uiff(P;Q),
btrue: tt,
sq_type: SQType(T),
uimplies: b supposing a,
so_apply: x[s],
ycomb: Y,
strongwellfounded: SWellFounded(R[x; y]),
iterated_classrel: iterated_classrel(es;S;A;f;init;X;e;v),
es-locl: (e <loc e'),
trans: Trans(T;x,y.E[x; y]),
decidable: Dec(P),
subtype: S 
T,
!hyp_hide: x
Lemmas :
assert_of_bnot,
eqff_to_assert,
uiff_transitivity,
eqtt_to_assert,
bool_cases,
bnot_wf,
assert_wf,
assert_elim,
es-locl-first,
btrue_neq_bfalse,
es-first_wf,
bag_wf,
Id_wf,
eclass_wf,
event-ordering+_wf,
single-valued-classrel_wf,
sq_stable_wf,
trans_wf,
false_wf,
bool_subtype_base,
bool_wf,
subtype_base_sq,
es-pred_wf,
es-E_wf,
event-ordering+_inc,
es-loc_wf,
single-valued-bag_wf,
iterated_classrel_wf,
squash_wf,
sq_stable__equal,
sq_stable__all,
equal_wf,
not_wf,
all_wf,
classrel_wf,
es-le_wf,
es-locl_wf,
and_wf,
exists_wf,
sq_stable__and,
es-causl_wf,
le_wf,
nat_wf,
less_than_wf,
ge_wf,
nat_properties,
es-causl-swellfnd,
es-loc-pred,
es-locl-trichotomy,
es-le_weakening,
es-locl_transitivity1,
es-pred-locl,
es-causl_weakening,
es-pred-causl,
member_wf,
implies-es-le-pred,
decidable__exists-classrel-between3,
decidable__exists-single-valued-classrel,
iterated_classrel-single-val,
es-causl_irreflexivity,
es-causle_weakening,
es-causl_transitivity2,
es-locl_irreflexivity,
es-le_weakening_eq,
es-locl_transitivity2,
es-pred_property
\mforall{}[Info,A,S:Type]. \mforall{}[init:Id {}\mrightarrow{} bag(S)]. \mforall{}[f:A {}\mrightarrow{} S {}\mrightarrow{} S]. \mforall{}[X:EClass(A)].
\mforall{}es:EO+(Info)
(single-valued-classrel(es;X;A)
{}\mRightarrow{} (\mforall{}[R:S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}]
((\mforall{}x,y:S. SqStable(R[x;y]))
{}\mRightarrow{} Trans(S;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}e1,e2:E.
((\mforall{}a:A. \mforall{}e:E.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} (\mforall{}s:S. (iterated\_classrel(es;S;A;f;init;X;pred(e);s) {}\mRightarrow{} R[s;f a s]))))
{}\mRightarrow{} single-valued-bag(init loc(e1);S)
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} (\mforall{}v1,v2:S.
(iterated\_classrel(es;S;A;f;init;X;e1;v1)
{}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e2;v2)
{}\mRightarrow{} (((\mexists{}e:E. ((e1 <loc e) \mwedge{} e \mleq{}loc e2 \mwedge{} (\mexists{}a:A. a \mmember{} X(e)))) {}\mRightarrow{} R[v1;v2])
\mwedge{} ((\mforall{}e:E. ((e1 <loc e) {}\mRightarrow{} e \mleq{}loc e2 {}\mRightarrow{} (\mforall{}a:A. (\mneg{}a \mmember{} X(e)))))
{}\mRightarrow{} (v1 = v2))))))))))
Date html generated:
2012_02_20-PM-02_47_56
Last ObjectModification:
2012_01_29-PM-07_08_35
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