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Step * 1 1 4 of Lemma cs-ref-map3-decided


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. filter(λx.cs-is-committed(x);L) = [] ∈ (consensus-state3(V) List)
6. ¬(filter(λx.cs-is-considering(x);L) = [] ∈ (consensus-state3(V) List))
7. PREDECIDED[cs-considered-val(hd(filter(λx.cs-is-considering(x);L)))] = Decided[v] ∈ consensus-state2(V)@i
8. ∀[i:ℕ||L||]. (¬↑cs-is-committed(L[i]))
⊢ (COMMITED[v] ∈ L)
BY
{ (RepUR ``cs-ambivalent cs-decided cs-predecided consensus-state2`` -2 THEN Auto) }

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Latex:



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.  filter(\mlambda{}x.cs-is-committed(x);L)  =  []
6.  \mneg{}(filter(\mlambda{}x.cs-is-considering(x);L)  =  [])
7.  PREDECIDED[cs-considered-val(hd(filter(\mlambda{}x.cs-is-considering(x);L)))]  =  Decided[v]@i
8.  \mforall{}[i:\mBbbN{}||L||].  (\mneg{}\muparrow{}cs-is-committed(L[i]))
\mvdash{}  (COMMITED[v]  \mmember{}  L)


By

(RepUR  ``cs-ambivalent  cs-decided  cs-predecided  consensus-state2``  -2  THEN  Auto)

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