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Step * 3 1 3 3 3 of Lemma consensus-refinement2

.....wf..... 
1. V : Type
2. ∀x,y:ts-reachable(consensus-ts3(V)).
     ((x ts-rel(consensus-ts3(V)) y) ⇒ (((λL.cs-ref-map3(L)) x) (ts-rel(consensus-ts2(V))^*) ((λL.cs-ref-map3(L)) y)))
3. x : ts-reachable(consensus-ts2(V))@i
4. v : V@i
5. x = Decided[v] ∈ consensus-state2(V)@i
6. 1 = 1 ∈ ℤ
7. i : ℕ1
⊢ (∀j:ℕ1. ((¬(j = i ∈ ℤ)) ⇒ ([CONSIDERING[v]][j] = [INITIAL][j] ∈ consensus-state3(V))))
  ∧ ((([INITIAL][i] = INITIAL ∈ consensus-state3(V))
    ∧ (([CONSIDERING[v]][i] = WITHDRAWN ∈ consensus-state3(V))
      ∨ (∃v@0:V
          (([CONSIDERING[v]][i] = CONSIDERING[v@0] ∈ consensus-state3(V))
          ∧ (∀j:ℕi
               (([INITIAL][j] = WITHDRAWN ∈ consensus-state3(V))
               ∨ ([INITIAL][j] = CONSIDERING[v@0] ∈ consensus-state3(V))
               ∨ ([INITIAL][j] = COMMITED[v@0] ∈ consensus-state3(V))))))))
    ∨ (∃v@0:V
        (([INITIAL][i] = CONSIDERING[v@0] ∈ consensus-state3(V))
        ∧ (([CONSIDERING[v]][i] = COMMITED[v@0] ∈ consensus-state3(V))
          ∨ ([CONSIDERING[v]][i] = WITHDRAWN ∈ consensus-state3(V)))))) ∈ ℙ
BY
{ (Auto THEN Reduce 0 THEN Auto') }

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Latex:


.....wf..... 
1.  V  :  Type
2.  \mforall{}x,y:ts-reachable(consensus-ts3(V)).
          ((x  ts-rel(consensus-ts3(V))  y)
          {}\mRightarrow{}  (((\mlambda{}L.cs-ref-map3(L))  x)  (ts-rel(consensus-ts2(V))\^{}*)  ((\mlambda{}L.cs-ref-map3(L))  y)))
3.  x  :  ts-reachable(consensus-ts2(V))@i
4.  v  :  V@i
5.  x  =  Decided[v]@i
6.  1  =  1
7.  i  :  \mBbbN{}1
\mvdash{}  (\mforall{}j:\mBbbN{}1.  ((\mneg{}(j  =  i))  {}\mRightarrow{}  ([CONSIDERING[v]][j]  =  [INITIAL][j])))
    \mwedge{}  ((([INITIAL][i]  =  INITIAL)
        \mwedge{}  (([CONSIDERING[v]][i]  =  WITHDRAWN)
            \mvee{}  (\mexists{}v@0:V
                    (([CONSIDERING[v]][i]  =  CONSIDERING[v@0])
                    \mwedge{}  (\mforall{}j:\mBbbN{}i
                              (([INITIAL][j]  =  WITHDRAWN)
                              \mvee{}  ([INITIAL][j]  =  CONSIDERING[v@0])
                              \mvee{}  ([INITIAL][j]  =  COMMITED[v@0])))))))
        \mvee{}  (\mexists{}v@0:V
                (([INITIAL][i]  =  CONSIDERING[v@0])
                \mwedge{}  (([CONSIDERING[v]][i]  =  COMMITED[v@0])  \mvee{}  ([CONSIDERING[v]][i]  =  WITHDRAWN)))))  \mmember{}  \mBbbP{}


By

(Auto  THEN  Reduce  0  THEN  Auto')

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