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 ─→ ℙ. ∀e1,e2:E. ∀v1,v2:S. ∀a:A.
(single-valued-classrel(es;X;A)
⇒ single-valued-bag(init loc(e1);S)
⇒ (∀a:A. ∀s:S. ∀e:E.
(e1 ≤loc e ⇒ e ≤loc e2 ⇒ a ∈ X(e) ⇒ prior-iterated-classrel(es;A;S;s;X;f;init;e) ⇒ R[a;s;f a s]))
⇒ (∀a1,a2:A. ∀s1,s2:S. ∀e,e':E.
(e1 ≤loc e
⇒ (e <loc e')
⇒ e' ≤loc e2
⇒ a1 ∈ X(e)
⇒ prior-iterated-classrel(es;A;S;s1;X;f;init;e)
⇒ a2 ∈ X(e')
⇒ iterated-classrel(es;S;A;f;init;X;pred(e');s2)
⇒ R[a1;s1;s2]
⇒ R[a1;s1;f a2 s2]))
⇒ e1 ≤loc e2
⇒ a ∈ X(e1)
⇒ prior-iterated-classrel(es;A;S;v1;X;f;init;e1)
⇒ iterated-classrel(es;S;A;f;init;X;e2;v2)
⇒ R[a;v1;v2])
Proof
Definitions occuring in Statement :
prior-iterated-classrel: prior-iterated-classrel(es;A;S;s;X;f;init;e),
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,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
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)
Lemmas :
es-causl-swellfnd,
less_than_wf,
decidable__lt,
subtract_wf,
false_wf,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
add-commutes,
less-iff-le,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
es-causl_wf,
es-first_wf2,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
iterated-classrel_wf,
prior-iterated-classrel_wf,
classrel_wf,
es-le_wf,
all_wf,
es-locl_wf,
es-pred_wf,
es-locl-first,
assert_elim,
btrue_neq_bfalse,
assert_wf,
single-valued-bag_wf,
es-loc_wf,
single-valued-classrel_wf,
nat_wf,
set_wf,
primrec-wf2,
zero-le-nat,
le_wf,
add-mul-special,
zero-mul,
add-zero,
es-E_wf,
event-ordering+_subtype,
eclass_wf,
event-ordering+_wf,
Id_wf,
bag_wf,
es-le-first,
and_wf,
equal_wf,
ifthenelse_wf,
es-le-loc,
bag-member_wf,
es-le-self,
not_wf,
not_assert_elim,
es-pred-locl,
es-causl_weakening,
es-locl_transitivity1,
es-le_weakening,
es-le-pred,
iff_weakening_equal,
member_wf,
squash_wf,
true_wf,
event_ordering_wf,
iterated-classrel-single-val,
es-pred-loc-base
\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{}a:A. \mforall{}s:S. \mforall{}e:E.
(e1 \mleq{}loc e
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} prior-iterated-classrel(es;A;S;s;X;f;init;e)
{}\mRightarrow{} R[a;s;f a s]))
{}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s1,s2:S. \mforall{}e,e':E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e')
{}\mRightarrow{} e' \mleq{}loc e2
{}\mRightarrow{} a1 \mmember{} X(e)
{}\mRightarrow{} prior-iterated-classrel(es;A;S;s1;X;f;init;e)
{}\mRightarrow{} a2 \mmember{} X(e')
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e');s2)
{}\mRightarrow{} R[a1;s1;s2]
{}\mRightarrow{} R[a1;s1;f a2 s2]))
{}\mRightarrow{} e1 \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e1)
{}\mRightarrow{} prior-iterated-classrel(es;A;S;v1;X;f;init;e1)
{}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;e2;v2)
{}\mRightarrow{} R[a;v1;v2])
Date html generated:
2015_07_17-PM-00_28_39
Last ObjectModification:
2015_02_04-PM-05_33_58
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