Nuprl Lemma : Memory-loc-class-trans-refl

[Info,B,A:Type].
  ∀R:B ─→ B ─→ ℙ. ∀f:Id ─→ 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 ∈ Memory-loc-class(f;init;X)(e)  R[s;f loc(e) s]))))
     single-valued-classrel(es;X;A)
     single-valued-bag(init loc(e1);B)
     v1 ∈ Memory-loc-class(f;init;X)(e1)
     v2 ∈ Memory-loc-class(f;init;X)(e2)
     e1 ≤loc e2 
     R[v1;v2])


Proof




Definitions occuring in Statement :  Memory-loc-class: Memory-loc-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]) 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:  Q apply: a function: x:A ─→ B[x] universe: Type single-valued-bag: single-valued-bag(b;T) bag: bag(T)
Lemmas :  Memory-classrel-loc classrel_wf Memory-class_wf es-loc_wf Memory-class-trans-refl es-le-loc and_wf equal_wf Id_wf eclass_wf es-E_wf event-ordering+_subtype event-ordering+_wf es-locl_wf es-le_wf Memory-loc-class_wf single-valued-bag_wf single-valued-classrel_wf all_wf sq_stable_wf trans_wf refl_wf bag_wf

Latex:
\mforall{}[Info,B,A:Type].
    \mforall{}R:B  {}\mrightarrow{}  B  {}\mrightarrow{}  \mBbbP{}.  \mforall{}f:Id  {}\mrightarrow{}  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  \mleq{}loc  e 
                    {}\mRightarrow{}  (e  <loc  e2)
                    {}\mRightarrow{}  a  \mmember{}  X(e)
                    {}\mRightarrow{}  (\mforall{}s:B.  (s  \mmember{}  Memory-loc-class(f;init;X)(e)  {}\mRightarrow{}  R[s;f  loc(e)  a  s]))))
        {}\mRightarrow{}  single-valued-classrel(es;X;A)
        {}\mRightarrow{}  single-valued-bag(init  loc(e1);B)
        {}\mRightarrow{}  v1  \mmember{}  Memory-loc-class(f;init;X)(e1)
        {}\mRightarrow{}  v2  \mmember{}  Memory-loc-class(f;init;X)(e2)
        {}\mRightarrow{}  e1  \mleq{}loc  e2 
        {}\mRightarrow{}  R[v1;v2])



Date html generated: 2015_07_22-PM-00_14_08
Last ObjectModification: 2015_01_28-AM-10_46_16

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