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Step * of Lemma State-loc-comb-trans1

∀[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.
    (Trans(B;x,y.R[x;y])
    ⇒ (∀a:A. ∀e:E.
          ((e1 <loc e)
          ⇒ e ≤loc e2 
          ⇒ a ∈ X(e)
          ⇒ (∀s:B. (s ∈ State-loc-comb(init;f;X)(pred(e)) ⇒ R[s;f loc(e) a s]))))
    ⇒ single-valued-classrel(es;X;A)
    ⇒ single-valued-bag(init loc(e1);B)
    ⇒ v1 ∈ State-loc-comb(init;f;X)(e1)
    ⇒ v2 ∈ State-loc-comb(init;f;X)(e2)
    ⇒ (e1 <loc e2)
    ⇒ (∃e:E. ((e1 <loc e) ∧ e ≤loc e2  ∧ (∃a:A. a ∈ X(e))))
    ⇒ R[v1;v2])
BY
{ ((UnivCD THENA Auto)
   THEN (RWO "State-loc-comb-classrel-non-loc" (-3) THENA Auto)
   THEN (RWO "State-loc-comb-classrel-non-loc" (-4) THENA Auto)
   THEN InstLemma `State-comb-trans1` [⌈Info⌉;⌈B⌉;⌈A⌉;⌈R⌉;⌈f loc(e1)⌉;⌈init⌉;⌈X⌉;⌈es⌉;⌈e1⌉;⌈e2⌉;⌈v1⌉;⌈v2⌉]⋅
   THEN Auto
   THEN InstHyp [⌈a⌉;⌈e⌉;⌈s⌉] (-14)⋅
   THEN Auto
   THEN RWO "State-loc-comb-classrel-non-loc" 0
   THEN Auto) }

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Latex:

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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.
        (Trans(B;x,y.R[x;y])
        {}\mRightarrow{}  (\mforall{}a:A.  \mforall{}e:E.
                    ((e1  <loc  e)
                    {}\mRightarrow{}  e  \mleq{}loc  e2 
                    {}\mRightarrow{}  a  \mmember{}  X(e)
                    {}\mRightarrow{}  (\mforall{}s:B.  (s  \mmember{}  State-loc-comb(init;f;X)(pred(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{}  State-loc-comb(init;f;X)(e1)
        {}\mRightarrow{}  v2  \mmember{}  State-loc-comb(init;f;X)(e2)
        {}\mRightarrow{}  (e1  <loc  e2)
        {}\mRightarrow{}  (\mexists{}e:E.  ((e1  <loc  e)  \mwedge{}  e  \mleq{}loc  e2    \mwedge{}  (\mexists{}a:A.  a  \mmember{}  X(e))))
        {}\mRightarrow{}  R[v1;v2])


By

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Latex:
((UnivCD  THENA  Auto)
  THEN  (RWO  "State-loc-comb-classrel-non-loc"  (-3)  THENA  Auto)
  THEN  (RWO  "State-loc-comb-classrel-non-loc"  (-4)  THENA  Auto)
  THEN  InstLemma  `State-comb-trans1`  [\mkleeneopen{}Info\mkleeneclose{};\mkleeneopen{}B\mkleeneclose{};\mkleeneopen{}A\mkleeneclose{};\mkleeneopen{}R\mkleeneclose{};\mkleeneopen{}f  loc(e1)\mkleeneclose{};\mkleeneopen{}init\mkleeneclose{};\mkleeneopen{}X\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e1\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{};\mkleeneopen{}v1\mkleeneclose{};
  \mkleeneopen{}v2\mkleeneclose{}]\mcdot{}
  THEN  Auto
  THEN  InstHyp  [\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}s\mkleeneclose{}]  (-14)\mcdot{}
  THEN  Auto
  THEN  RWO  "State-loc-comb-classrel-non-loc"  0
  THEN  Auto)

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