Nuprl Lemma : State-comb-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 s]))
     (∀a1,a2:A. ∀s1,s2:B. ∀e,e':E.
          (e1 ≤loc 
           (e <loc e')
           e' ≤loc e2 
           a1 ∈ X(e)
           s1 ∈ Memory-class(f;init;X)(e)
           a2 ∈ X(e')
           s2 ∈ State-comb(init;f;X)(pred(e'))
           R[a1;s1;s2]
           R[a1;s1;f a2 s2]))
     (∀s1,s2:B. ∀a:A.  SqStable(R[a;s1;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 ∈ State-comb(init;f;X)(e2)
     R[a;v1;v2])


Proof




Definitions occuring in Statement :  State-comb: State-comb(init;f;X) 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-pred: pred(e) es-loc: loc(e) es-E: E Id: Id sq_stable: SqStable(P) uall: [x:A]. B[x] prop: so_apply: x[s1;s2;s3] 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 :  classrel_wf State-comb_wf Memory-class_wf es-le_wf event-ordering+_subtype single-valued-bag_wf es-loc_wf single-valued-classrel_wf all_wf sq_stable_wf es-E_wf es-locl_wf es-pred_wf es-locl-first assert_elim btrue_neq_bfalse assert_wf es-first_wf2 event-ordering+_wf eclass_wf Id_wf bag_wf iterated_classrel_mem2 prior_iterated_classrel_wf iterated_classrel_wf State-comb-classrel Memory-classrel not_wf 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  \mleq{}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'  \mleq{}loc  e2 
                    {}\mRightarrow{}  a1  \mmember{}  X(e)
                    {}\mRightarrow{}  s1  \mmember{}  Memory-class(f;init;X)(e)
                    {}\mRightarrow{}  a2  \mmember{}  X(e')
                    {}\mRightarrow{}  s2  \mmember{}  State-comb(init;f;X)(pred(e'))
                    {}\mRightarrow{}  R[a1;s1;s2]
                    {}\mRightarrow{}  R[a1;s1;f  a2  s2]))
        {}\mRightarrow{}  (\mforall{}s1,s2:B.  \mforall{}a:A.    SqStable(R[a;s1;s2]))
        {}\mRightarrow{}  single-valued-classrel(es;X;A)
        {}\mRightarrow{}  single-valued-bag(init  loc(e1);B)
        {}\mRightarrow{}  e1  \mleq{}loc  e2 
        {}\mRightarrow{}  a  \mmember{}  X(e1)
        {}\mRightarrow{}  v1  \mmember{}  Memory-class(f;init;X)(e1)
        {}\mRightarrow{}  v2  \mmember{}  State-comb(init;f;X)(e2)
        {}\mRightarrow{}  R[a;v1;v2])



Date html generated: 2015_07_22-PM-00_22_48
Last ObjectModification: 2015_01_28-AM-10_12_57

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