Nuprl Lemma : iterated_classrel_mem
[Info,A,S:Type]. [init:Id 
bag(S)]. [f:A S S]. [X:EClass(A)].
es:EO+(Info). R:A S S 
. e1,e2:E. v1,v2:S. a:A.
(single-valued-classrel(es;X;A)
single-valued-bag(init loc(e1);S)
(s1,s2:S. a:A. SqStable(R[a;s1;s2]))
(a1,a2:A. s:S. e,e':E.
(e1 
loc e (e <loc e') e' loc e2 a1 
X(e) s X*(f,init,e) a2 X(e') R[a1;s;f a2 s]))
(a1,a2:A. s1,s2:S. e1,e2:E.
((e1 <loc e2)
a1 X(e1)
s1 X*(f,init,e1)
a2 X(e2)
s2 X*(f,init,pred(e2))
R[a1;s1;s2]
R[a1;s1;f a2 s2]))
(e1 <loc e2)
a X(e1)
v1 X*(f,init,e1)
v2 X*(f,init,e2)
(((
e:E. ((e1 <loc e) 
e loc e2 (a:A. a X(e)))) R[a;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-pred: pred(e),
es-loc: loc(e),
es-E: E,
Id: Id,
sq_stable: SqStable(P),
uall: [x:A]. B[x],
prop: ,
so_apply: x[s1;s2;s3],
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],
prop: ,
implies: P Q,
sq_stable: SqStable(P),
so_apply: x[s1;s2;s3],
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,
ifthenelse: if b then t else f fi ,
so_lambda: 
x.t[x],
assert: 
b,
bfalse: ff,
so_lambda: x y.t[x; y],
rev_implies: P 
Q,
iff: P Q,
es-le: e loc e' ,
or: P 
Q,
guard: {T},
cand: A c B,
label: ...$L... t,
es-locl: (e <loc e'),
iterated_classrel: v X*(f,init,e),
strongwellfounded: SWellFounded(R[x; y]),
ycomb: Y,
so_apply: x[s],
uimplies: b supposing a,
sq_type: SQType(T),
so_apply: x[s1;s2],
btrue: tt,
uiff: uiff(P;Q),
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,
es-E_wf,
es-pred_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
false_wf,
sq_stable_wf,
single-valued-bag_wf,
es-loc_wf,
single-valued-classrel_wf,
eclass_wf,
event-ordering+_wf,
Id_wf,
bag_wf,
es-first_wf,
btrue_neq_bfalse,
es-locl-first,
assert_elim,
assert_wf,
bnot_wf,
es-locl-trichotomy,
es-loc-pred,
bool_cases,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
es-pred-causl,
es-causl_weakening,
es-pred-locl,
es-locl_transitivity1,
es-le_weakening,
decidable__exists-classrel-between3-sv,
member_wf,
implies-es-le-pred,
iterated_classrel-single-val,
es-pred_property,
es-locl_transitivity2,
es-le_weakening_eq,
es-locl_irreflexivity,
es-causl_transitivity2,
es-causle_weakening,
es-causle_weakening_eq,
es-causl_irreflexivity,
es-causle_wf
\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). \mforall{}R:A {}\mrightarrow{} S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}. \mforall{}e1,e2:E. \mforall{}v1,v2:S. \mforall{}a:A.
(single-valued-classrel(es;X;A)
{}\mRightarrow{} single-valued-bag(init loc(e1);S)
{}\mRightarrow{} (\mforall{}s1,s2:S. \mforall{}a:A. SqStable(R[a;s1;s2]))
{}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s:S. \mforall{}e,e':E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e')
{}\mRightarrow{} e' \mleq{}loc e2
{}\mRightarrow{} a1 \mmember{} X(e)
{}\mRightarrow{} s \mmember{} X*(f,init,e)
{}\mRightarrow{} a2 \mmember{} X(e')
{}\mRightarrow{} R[a1;s;f a2 s]))
{}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s1,s2:S. \mforall{}e1,e2:E.
((e1 <loc e2)
{}\mRightarrow{} a1 \mmember{} X(e1)
{}\mRightarrow{} s1 \mmember{} X*(f,init,e1)
{}\mRightarrow{} a2 \mmember{} X(e2)
{}\mRightarrow{} s2 \mmember{} X*(f,init,pred(e2))
{}\mRightarrow{} R[a1;s1;s2]
{}\mRightarrow{} R[a1;s1;f a2 s2]))
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} a \mmember{} X(e1)
{}\mRightarrow{} 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[a;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_21_31
Last ObjectModification:
2012_01_20-AM-02_00_47
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