Nuprl Lemma : iterated_classrel_trans2
[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])
(a:A. e:E. (a 
X(e) (s:S. R[s;f a s])))
(e1,e2:E.
(single-valued-bag(init loc(e1);S)
(e1 <loc e2)
(v1,v2:S.
(v1 X*(f,init,e1)
v2 X*(f,init,e2)
(((
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: v X*(f,init,e),
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],
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 :
uall: [x:A]. B[x],
all: x:A. B[x],
implies: P Q,
prop: ,
sq_stable: SqStable(P),
so_apply: x[s1;s2],
and: P Q,
exists: x:A. B[x],
not: A,
member: t T,
nat: 
,
ge: i 
j ,
le: A B,
false: False,
squash: 
T,
true: True,
so_lambda: 
x.t[x],
so_lambda: x y.t[x; y],
cand: A c B,
es-le: e loc e' ,
or: P 
Q,
guard: {T},
iterated_classrel: v X*(f,init,e),
strongwellfounded: SWellFounded(R[x; y]),
ycomb: Y,
so_apply: x[s],
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
uiff: uiff(P;Q),
es-locl: (e <loc e'),
iff: P 
Q,
rev_implies: P Q,
trans: Trans(T;x,y.E[x; y]),
decidable: Dec(P),
subtype: S 
T,
!hyp_hide: x
Lemmas :
es-causl-swellfnd,
nat_properties,
ge_wf,
less_than_wf,
nat_wf,
le_wf,
es-causl_wf,
sq_stable__and,
exists_wf,
and_wf,
es-locl_wf,
es-le_wf,
classrel_wf,
all_wf,
not_wf,
equal_wf,
sq_stable__all,
sq_stable__equal,
squash_wf,
iterated_classrel_wf,
event-ordering+_inc,
single-valued-bag_wf,
es-loc_wf,
es-E_wf,
trans_wf,
sq_stable_wf,
single-valued-classrel_wf,
event-ordering+_wf,
eclass_wf,
Id_wf,
bag_wf,
es-first_wf,
btrue_neq_bfalse,
es-locl-first,
assert_elim,
bool_wf,
assert_wf,
bnot_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
es-locl-trichotomy,
es-pred_wf,
es-loc-pred,
es-pred-causl,
es-causl_weakening,
es-le-pred,
member_wf,
decidable__exists-classrel-between3,
decidable__exists-single-valued-classrel,
es-pred-locl,
es-locl_transitivity1,
es-le_weakening,
iterated_classrel-single-val,
es-pred_property,
es-locl_transitivity2,
es-le_weakening_eq,
es-locl_irreflexivity,
es-causl_transitivity2,
es-causle_weakening,
es-causl_irreflexivity
\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{}a:A. \mforall{}e:E. (a \mmember{} X(e) {}\mRightarrow{} (\mforall{}s:S. R[s;f a s])))
{}\mRightarrow{} (\mforall{}e1,e2:E.
(single-valued-bag(init loc(e1);S)
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} (\mforall{}v1,v2:S.
(v1 \mmember{} X*(f,init,e1)
{}\mRightarrow{} v2 \mmember{} X*(f,init,e2)
{}\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_01_23-PM-12_20_57
Last ObjectModification:
2012_01_10-PM-07_43_23
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