Nuprl Lemma : Memory-class-trans1
∀[Info,B,A:Type].
∀R:B ─→ B ─→ ℙ. ∀f:A ─→ B ─→ B. ∀init:Id ─→ bag(B). ∀X:EClass(A). ∀es:EO+(Info). ∀e1,e2:E. ∀v1,v2:B.
(Trans(B;x,y.R[x;y])
⇒ (∀a:A. ∀e:E. (e1 ≤loc e ⇒ (e <loc e2) ⇒ a ∈ X(e) ⇒ (∀s:B. (s ∈ Memory-class(f;init;X)(e) ⇒ R[s;f a s]))))
⇒ single-valued-classrel(es;X;A)
⇒ single-valued-bag(init loc(e1);B)
⇒ v1 ∈ Memory-class(f;init;X)(e1)
⇒ v2 ∈ Memory-class(f;init;X)(e2)
⇒ (e1 <loc e2)
⇒ (∃e:E. (e1 ≤loc e ∧ (e <loc e2) ∧ (∃a:A. a ∈ X(e))))
⇒ R[v1;v2])
Proof
Definitions occuring in Statement :
Memory-class: Memory-class(f;init;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-loc: loc(e),
es-E: E,
Id: Id,
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
single-valued-bag: single-valued-bag(b;T),
bag: bag(T)
Lemmas :
decidable__exists-single-valued-classrel,
single-valued-bag_wf,
bag_wf,
Id_wf,
equal_wf,
and_wf,
es-loc-pred,
event-ordering+_subtype,
es-pred_wf,
sq_stable__single-valued-iterated-classrel,
es-causl_weakening,
es-pred-locl,
es-pred_property,
exists_wf,
or_wf,
es-loc_wf,
bag-member_wf,
not_wf,
all_wf,
classrel_wf,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
bool_wf,
es-first_wf2,
iff_weakening_equal,
event_ordering_wf,
es-E_wf,
true_wf,
squash_wf,
iterated-classrel-single-val,
iterated-classrel_wf,
iterated_classrel_wf,
assert_wf,
Memory-classrel,
Memory-class_wf,
sq_stable__classrel,
es-le-self,
decidable__exists-classrel-between3-sv,
es-locl_wf,
es-le_wf,
btrue_neq_bfalse,
assert_elim,
es-locl-first,
Memory-classrel2,
es-locl_transitivity1,
es-le_weakening,
iterated-classrel-trans,
es-causle_wf,
es-causle_weakening_eq,
eclass_wf,
event-ordering+_wf,
es-causl_wf,
es-causl_irreflexivity,
es-causle_weakening,
es-causl_transitivity2,
es-le-pred,
es-locl-trichotomy,
es-locl_transitivity2,
es-pred-loc-base,
es-pred-locl-implies-le,
es-locl-pred
Latex:
\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.
(Trans(B;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e2)
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} (\mforall{}s:B. (s \mmember{} Memory-class(f;init;X)(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{} Memory-class(f;init;X)(e1)
{}\mRightarrow{} v2 \mmember{} Memory-class(f;init;X)(e2)
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} (\mexists{}e:E. (e1 \mleq{}loc e \mwedge{} (e <loc e2) \mwedge{} (\mexists{}a:A. a \mmember{} X(e))))
{}\mRightarrow{} R[v1;v2])
Date html generated:
2015_07_22-PM-00_13_34
Last ObjectModification:
2015_07_16-AM-09_39_40
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