Nuprl Lemma : Memory-class-progress

[Info,B,A:Type].
  ∀R:B ─→ B ─→ ℙ. ∀P:A ─→ B ─→ ℙ. ∀f:A ─→ B ─→ B. ∀init:Id ─→ bag(B). ∀X:EClass(A). ∀es:EO+(Info). ∀e1,e2:E. ∀v1,v2:B.
    ((∀a:A. ∀s:B.  Dec(P[a;s]))
     Trans(B;x,y.R[x;y])
     (∀s1,s2:B.  SqStable(R[s1;s2]))
     (∀a:A. ∀e:E. ∀s:B.
          (e1 ≤loc 
           (e <loc e2)
           a ∈ X(e)
           s ∈ Memory-class(f;init;X)(e)
           ((P[a;s]  R[s;f s]) ∧ ((¬P[a;s])  (s (f s) ∈ B)))))
     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. ∃a:A. ∃s:B. (e1 ≤loc e  ∧ (e <loc e2) ∧ s ∈ Memory-class(f;init;X)(e) ∧ a ∈ X(e) ∧ P[a;s]))
     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]) sq_stable: SqStable(P) decidable: Dec(P) uall: [x:A]. B[x] prop: so_apply: x[s1;s2] all: x:A. B[x] exists: x:A. B[x] not: ¬A implies:  Q and: P ∧ Q apply: a function: x:A ─→ B[x] universe: Type equal: t ∈ T single-valued-bag: single-valued-bag(b;T) bag: bag(T)
Lemmas :  decidable__exists-single-valued-classrel subtype_base_sq bool_wf bool_subtype_base iff_imp_equal_bool es-first_wf2 event-ordering+_subtype btrue_wf assert_wf true_wf classrel_wf all_wf not_wf bag-member_wf es-loc_wf or_wf exists_wf es-pred_property es-pred-locl es-causl_weakening es-le-self Memory-classrel iterated_classrel_wf es-pred_wf iterated_classrel-single-val single-valued-bag_wf squash_wf bag_wf es-E_wf event_ordering_wf iff_weakening_equal es-le-loc not_assert_elim and_wf equal_wf assert_elim btrue_neq_bfalse es-causl_transitivity2 es-causle_weakening_eq es-causle_wf es-causle_weakening_locl es-causl_irreflexivity iterated_classrel_progress es-locl-first es-le_wf es-locl_wf es-loc-pred es-le_weakening es-locl_transitivity1 decidable__exists-iterated-classrel-between3-sv es-le-pred Id_wf Memory-class_wf event-ordering+_wf eclass_wf ifthenelse_wf es-le-first es-pred-loc-base es-locl-trichotomy es-locl_transitivity2 es-le_weakening_eq es-locl_irreflexivity es-causle_weakening

Latex:
\mforall{}[Info,B,A:Type].
    \mforall{}R:B  {}\mrightarrow{}  B  {}\mrightarrow{}  \mBbbP{}.  \mforall{}P:A  {}\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{}s:B.    Dec(P[a;s]))
        {}\mRightarrow{}  Trans(B;x,y.R[x;y])
        {}\mRightarrow{}  (\mforall{}s1,s2:B.    SqStable(R[s1;s2]))
        {}\mRightarrow{}  (\mforall{}a:A.  \mforall{}e:E.  \mforall{}s:B.
                    (e1  \mleq{}loc  e 
                    {}\mRightarrow{}  (e  <loc  e2)
                    {}\mRightarrow{}  a  \mmember{}  X(e)
                    {}\mRightarrow{}  s  \mmember{}  Memory-class(f;init;X)(e)
                    {}\mRightarrow{}  ((P[a;s]  {}\mRightarrow{}  R[s;f  a  s])  \mwedge{}  ((\mneg{}P[a;s])  {}\mRightarrow{}  (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
                  \mexists{}a:A.  \mexists{}s:B.  (e1  \mleq{}loc  e    \mwedge{}  (e  <loc  e2)  \mwedge{}  s  \mmember{}  Memory-class(f;init;X)(e)  \mwedge{}  a  \mmember{}  X(e)  \mwedge{}  P[a;s]))
        {}\mRightarrow{}  R[v1;v2])



Date html generated: 2015_07_22-PM-00_13_50
Last ObjectModification: 2015_02_04-PM-04_44_15

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