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Step * 1 3 2 of Lemma cs-ref-map3-predecided

.....falsecase..... 
1. [V] : Type
2. L : consensus-state3(V) List@i
3. v : V@i
4. ∀[v':V]. v' = v ∈ V supposing (CONSIDERING[v'] ∈ L) ∨ (COMMITED[v'] ∈ L) 
   supposing (CONSIDERING[v] ∈ L) ∨ (COMMITED[v] ∈ L)
5. ∀v:V. ((COMMITED[v] ∈ L) ⇐⇒ cs-ref-map3(L) = Decided[v] ∈ consensus-state2(V))
6. PREDECIDED[cs-considered-val(hd(filter(λx.cs-is-considering(x);L)))] = PREDECIDED[v] ∈ consensus-state2(V)@i
7. filter(λx.cs-is-committed(x);L) = [] ∈ (consensus-state3(V) List)
8. ¬(filter(λx.cs-is-considering(x);L) = [] ∈ (consensus-state3(V) List))
⊢ (CONSIDERING[v] ∈ L)
BY
{ (MoveToConcl (-3) THEN MoveToConcl (-1)) }

1
1. [V] : Type
2. L : consensus-state3(V) List@i
3. v : V@i
4. ∀[v':V]. v' = v ∈ V supposing (CONSIDERING[v'] ∈ L) ∨ (COMMITED[v'] ∈ L) 
   supposing (CONSIDERING[v] ∈ L) ∨ (COMMITED[v] ∈ L)
5. ∀v:V. ((COMMITED[v] ∈ L) ⇐⇒ cs-ref-map3(L) = Decided[v] ∈ consensus-state2(V))
6. filter(λx.cs-is-committed(x);L) = [] ∈ (consensus-state3(V) List)
⊢ (¬(filter(λx.cs-is-considering(x);L) = [] ∈ (consensus-state3(V) List)))
⇒ (PREDECIDED[cs-considered-val(hd(filter(λx.cs-is-considering(x);L)))] = PREDECIDED[v] ∈ consensus-state2(V))
⇒ (CONSIDERING[v] ∈ L)

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Latex:


.....falsecase..... 
1.  [V]  :  Type
2.  L  :  consensus-state3(V)  List@i
3.  v  :  V@i
4.  \mforall{}[v':V].  v'  =  v  supposing  (CONSIDERING[v']  \mmember{}  L)  \mvee{}  (COMMITED[v']  \mmember{}  L) 
      supposing  (CONSIDERING[v]  \mmember{}  L)  \mvee{}  (COMMITED[v]  \mmember{}  L)
5.  \mforall{}v:V.  ((COMMITED[v]  \mmember{}  L)  \mLeftarrow{}{}\mRightarrow{}  cs-ref-map3(L)  =  Decided[v])
6.  PREDECIDED[cs-considered-val(hd(filter(\mlambda{}x.cs-is-considering(x);L)))]  =  PREDECIDED[v]@i
7.  filter(\mlambda{}x.cs-is-committed(x);L)  =  []
8.  \mneg{}(filter(\mlambda{}x.cs-is-considering(x);L)  =  [])
\mvdash{}  (CONSIDERING[v]  \mmember{}  L)


By

(MoveToConcl  (-3)  THEN  MoveToConcl  (-1))

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