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Step * 3 1 3 3 2 of Lemma consensus-refinement2


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
⊢ (∀j:ℕ1. ((¬(j = 0 ∈ ℤ)) ⇒ ([CONSIDERING[v]][j] = [INITIAL][j] ∈ consensus-state3(V))))
∧ (((INITIAL = INITIAL ∈ consensus-state3(V))
  ∧ ((CONSIDERING[v] = WITHDRAWN ∈ consensus-state3(V))
    ∨ (∃v@0:V
        ((CONSIDERING[v] = CONSIDERING[v@0] ∈ consensus-state3(V))
        ∧ (∀j:ℕ0
             (([INITIAL][j] = WITHDRAWN ∈ consensus-state3(V))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 = CONSIDERING[v@0] ∈ consensus-state3(V))
      ∧ ((CONSIDERING[v] = COMMITED[v@0] ∈ consensus-state3(V)) ∨ (CONSIDERING[v] = WITHDRAWN ∈ consensus-state3(V))))))
BY
{ (OrLeft THEN Auto) }

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Latex:



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
\mvdash{}  (\mforall{}j:\mBbbN{}1.  ((\mneg{}(j  =  0))  {}\mRightarrow{}  ([CONSIDERING[v]][j]  =  [INITIAL][j])))
\mwedge{}  (((INITIAL  =  INITIAL)
    \mwedge{}  ((CONSIDERING[v]  =  WITHDRAWN)
        \mvee{}  (\mexists{}v@0:V
                ((CONSIDERING[v]  =  CONSIDERING[v@0])
                \mwedge{}  (\mforall{}j:\mBbbN{}0
                          (([INITIAL][j]  =  WITHDRAWN)
                          \mvee{}  ([INITIAL][j]  =  CONSIDERING[v@0])
                          \mvee{}  ([INITIAL][j]  =  COMMITED[v@0])))))))
    \mvee{}  (\mexists{}v@0:V
            ((INITIAL  =  CONSIDERING[v@0])
            \mwedge{}  ((CONSIDERING[v]  =  COMMITED[v@0])  \mvee{}  (CONSIDERING[v]  =  WITHDRAWN)))))


By

(OrLeft  THEN  Auto)

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