Nuprl Lemma : State-comb-trans-refl
[Info,B,A:Type]. [R:B 
B 
]. [f:A B B]. [init:Id bag(B)]. [X:EClass(A)]. [es:EO+(Info)]. [e1,e2:E].
v1,v2:B.
(Refl(B;x,y.R[x;y])
Trans(B;x,y.R[x;y])
(s1,s2:B. SqStable(R[s1;s2]))
(a:A. e:E.
((e1 <loc e) e 
loc e2 a 
X(e) (s:B. (s State-comb(init;f;X)(pred(e)) R[s;f a s]))))
single-valued-classrel(es;X;A)
single-valued-bag(init loc(e1);B)
v1 State-comb(init;f;X)(e1)
v2 State-comb(init;f;X)(e2)
e1 loc e2
R[v1;v2])
Proof not projected
Definitions occuring in Statement :
State-comb: State-comb(init;f;X),
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]),
refl: Refl(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],
implies: P Q,
apply: f a,
function: x:A B[x],
universe: Type,
single-valued-bag: single-valued-bag(b;T),
bag: bag(T)
Definitions :
uall: [x:A]. B[x],
prop: ,
all: x:A. B[x],
implies: P Q,
so_apply: x[s1;s2],
member: t T,
squash: 
T,
true: True,
and: P 
Q,
exists: 
x:A. B[x],
not: 
A,
assert: 
b,
bfalse: ff,
ifthenelse: if b then t else f fi ,
false: False,
cand: A c B,
so_lambda: 
x.t[x],
so_lambda: x y.t[x; y],
sq_stable: SqStable(P),
es-le: e loc e' ,
or: P 
Q,
rev_implies: P 
Q,
iff: P Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
decidable: Dec(P),
so_apply: x[s],
refl: Refl(T;x,y.E[x; y]),
subtype: S 
T
Lemmas :
iterated_classrel_trans3,
State-comb-classrel,
es-pred_wf,
iterated_classrel_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
false_wf,
classrel_wf,
es-le_wf,
es-locl_wf,
decidable__exists-classrel-between3-sv,
event-ordering+_inc,
exists_wf,
State-comb-single-val0,
State-comb_wf,
single-valued-bag_wf,
es-loc_wf,
single-valued-classrel_wf,
all_wf,
es-E_wf,
sq_stable_wf,
trans_wf,
refl_wf,
event-ordering+_wf,
eclass_wf,
Id_wf,
bag_wf,
es-locl-first,
and_wf,
equal_wf
\mforall{}[Info,B,A:Type]. \mforall{}[R:B {}\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.
(Refl(B;x,y.R[x;y])
{}\mRightarrow{} Trans(B;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}s1,s2:B. SqStable(R[s1;s2]))
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} (\mforall{}s:B. (s \mmember{} State-comb(init;f;X)(pred(e)) {}\mRightarrow{} R[s;f a s]))))
{}\mRightarrow{} single-valued-classrel(es;X;A)
{}\mRightarrow{} single-valued-bag(init loc(e1);B)
{}\mRightarrow{} v1 \mmember{} State-comb(init;f;X)(e1)
{}\mRightarrow{} v2 \mmember{} State-comb(init;f;X)(e2)
{}\mRightarrow{} e1 \mleq{}loc e2
{}\mRightarrow{} R[v1;v2])
Date html generated:
2012_02_20-PM-03_10_22
Last ObjectModification:
2012_02_17-AM-11_58_42
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