Nuprl Lemma : iterated_classrel_progress
[Info,A,S:Type]. [init:Id 
bag(S)]. [f:A S S]. [X:EClass(A)].
es:EO+(Info). R:S S 
. P:A S . e1,e2:E. v1,v2:S.
(single-valued-classrel(es;X;A)
(a:A. s:S. Dec(P[a;s]))
(x,y:S. SqStable(R[x;y]))
Trans(S;x,y.R[x;y])
(a:A. e:E. s:S.
((e1 <loc e)
e 
loc e2
a 
X(e)
s X*(f,init,pred(e))
((P[a;s] R[s;f a s]) 
((
P[a;s]) (s = (f a s))))))
single-valued-bag(init loc(e1);S)
(e1 <loc e2)
v1 X*(f,init,e1)
v2 X*(f,init,e2)
(((
e:E. s:S. ((e1 <loc e) e loc e2 s X*(f,init,pred(e)) (a:A. (a X(e) P[a;s])))) R[v1;v2])
((e:E. s:S. ((e1 <loc e) e loc e2 s X*(f,init,pred(e)) (a:A. ((a X(e)) 
(P[a;s])))))
(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,
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 :
uall: [x:A]. B[x],
all: x:A. B[x],
prop: ,
implies: P Q,
so_apply: x[s1;s2],
sq_stable: SqStable(P),
and: P Q,
not: A,
exists: x:A. B[x],
or: P Q,
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],
cand: A c B,
assert: 
b,
bfalse: ff,
so_lambda: x y.t[x; y],
es-le: e loc e' ,
guard: {T},
iff: P 
Q,
rev_implies: P Q,
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),
btrue: tt,
uiff: uiff(P;Q),
decidable: Dec(P),
trans: Trans(T;x,y.E[x; y]),
single-valued-classrel: single-valued-classrel(es;X;T),
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,
es-E_wf,
event-ordering+_inc,
es-locl_wf,
es-le_wf,
iterated_classrel_wf,
es-pred_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
false_wf,
classrel_wf,
all_wf,
or_wf,
not_wf,
equal_wf,
sq_stable__all,
sq_stable__equal,
squash_wf,
single-valued-bag_wf,
es-loc_wf,
and_wf,
trans_wf,
sq_stable_wf,
decidable_wf,
single-valued-classrel_wf,
eclass_wf,
event-ordering+_wf,
Id_wf,
bag_wf,
es-locl-first,
es-first_wf,
btrue_neq_bfalse,
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,
implies-es-le-pred,
member_wf,
decidable__exists-iterated-classrel-between3-sv,
iterated_classrel-single-val,
event_ordering_wf,
true_wf,
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). \mforall{}R:S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}. \mforall{}P:A {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}. \mforall{}e1,e2:E. \mforall{}v1,v2:S.
(single-valued-classrel(es;X;A)
{}\mRightarrow{} (\mforall{}a:A. \mforall{}s:S. Dec(P[a;s]))
{}\mRightarrow{} (\mforall{}x,y:S. SqStable(R[x;y]))
{}\mRightarrow{} Trans(S;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E. \mforall{}s:S.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} s \mmember{} X*(f,init,pred(e))
{}\mRightarrow{} ((P[a;s] {}\mRightarrow{} R[s;f a s]) \mwedge{} ((\mneg{}P[a;s]) {}\mRightarrow{} (s = (f a s))))))
{}\mRightarrow{} single-valued-bag(init loc(e1);S)
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} v1 \mmember{} X*(f,init,e1)
{}\mRightarrow{} v2 \mmember{} X*(f,init,e2)
{}\mRightarrow{} (((\mexists{}e:E
\mexists{}s:S. ((e1 <loc e) \mwedge{} e \mleq{}loc e2 \mwedge{} s \mmember{} X*(f,init,pred(e)) \mwedge{} (\mexists{}a:A. (a \mmember{} X(e) \mwedge{} P[a;s]))))
{}\mRightarrow{} R[v1;v2])
\mwedge{} ((\mforall{}e:E. \mforall{}s:S.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} s \mmember{} X*(f,init,pred(e))
{}\mRightarrow{} (\mforall{}a:A. ((\mneg{}a \mmember{} X(e)) \mvee{} (\mneg{}P[a;s])))))
{}\mRightarrow{} (v1 = v2))))
Date html generated:
2012_01_23-PM-12_21_19
Last ObjectModification:
2012_01_18-PM-05_36_00
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