Nuprl Lemma : Memory-class-mem
∀[Info,B,A:Type].
∀R:A ─→ B ─→ B ─→ ℙ. ∀f:A ─→ B ─→ B. ∀init:Id ─→ bag(B). ∀X:EClass(A). ∀es:EO+(Info). ∀e1,e2:E. ∀v1,v2:B. ∀a:A.
((∀a:A. ∀s:B. ∀e:E. (e1 ≤loc e ⇒ (e <loc e2) ⇒ a ∈ X(e) ⇒ s ∈ Memory-class(f;init;X)(e) ⇒ R[a;s;f a s]))
⇒ (∀a1,a2:A. ∀s1,s2:B. ∀e,e':E.
(e1 ≤loc e
⇒ (e <loc e')
⇒ (e' <loc e2)
⇒ a1 ∈ X(e)
⇒ s1 ∈ Memory-class(f;init;X)(e)
⇒ a2 ∈ X(e')
⇒ s2 ∈ Memory-class(f;init;X)(e')
⇒ R[a1;s1;s2]
⇒ R[a1;s1;f a2 s2]))
⇒ single-valued-classrel(es;X;A)
⇒ single-valued-bag(init loc(e1);B)
⇒ (e1 <loc e2)
⇒ a ∈ X(e1)
⇒ v1 ∈ Memory-class(f;init;X)(e1)
⇒ v2 ∈ Memory-class(f;init;X)(e2)
⇒ R[a;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,
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 :
Memory-classrel-single-val,
and_wf,
equal_wf,
Id_wf,
bag_wf,
single-valued-bag_wf,
es-locl-first,
assert_elim,
btrue_neq_bfalse,
es-first_wf2,
classrel_wf,
Memory-class_wf,
es-locl_wf,
event-ordering+_subtype,
es-loc_wf,
single-valued-classrel_wf,
all_wf,
es-E_wf,
es-le_wf,
event-ordering+_wf,
eclass_wf,
not_wf,
assert_wf,
iterated-classrel_wf,
es-pred_wf,
es-le-pred,
iterated-classrel-mem,
prior-iterated-classrel_wf,
es-pred-locl,
es-locl_transitivity1,
es-le-loc,
bag-member_wf
Latex:
\mforall{}[Info,B,A:Type].
\mforall{}R:A {}\mrightarrow{} 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. \mforall{}a:A.
((\mforall{}a:A. \mforall{}s:B. \mforall{}e:E.
(e1 \mleq{}loc e {}\mRightarrow{} (e <loc e2) {}\mRightarrow{} a \mmember{} X(e) {}\mRightarrow{} s \mmember{} Memory-class(f;init;X)(e) {}\mRightarrow{} R[a;s;f a s]))
{}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s1,s2:B. \mforall{}e,e':E.
(e1 \mleq{}loc e
{}\mRightarrow{} (e <loc e')
{}\mRightarrow{} (e' <loc e2)
{}\mRightarrow{} a1 \mmember{} X(e)
{}\mRightarrow{} s1 \mmember{} Memory-class(f;init;X)(e)
{}\mRightarrow{} a2 \mmember{} X(e')
{}\mRightarrow{} s2 \mmember{} Memory-class(f;init;X)(e')
{}\mRightarrow{} R[a1;s1;s2]
{}\mRightarrow{} R[a1;s1;f a2 s2]))
{}\mRightarrow{} single-valued-classrel(es;X;A)
{}\mRightarrow{} single-valued-bag(init loc(e1);B)
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} a \mmember{} X(e1)
{}\mRightarrow{} v1 \mmember{} Memory-class(f;init;X)(e1)
{}\mRightarrow{} v2 \mmember{} Memory-class(f;init;X)(e2)
{}\mRightarrow{} R[a;v1;v2])
Date html generated:
2015_07_22-PM-00_13_55
Last ObjectModification:
2015_01_28-AM-10_47_28
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