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


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. ∀v':V. (¬(COMMITED[v'] ∈ L))@i
7. (CONSIDERING[v] ∈ L)@i
8. filter(λx.cs-is-committed(x);L) = [] ∈ (consensus-state3(V) List)
9. filter(λx.cs-is-considering(x);L) = [] ∈ (consensus-state3(V) List)
⊢ uiff(True;∀[i:ℕ||L||]. (¬↑cs-is-considering(L[i])))
⇒ uiff(True;∀[i:ℕ||L||]. (¬↑cs-is-committed(L[i])))
⇒ (AMBIVALENT = PREDECIDED[v] ∈ consensus-state2(V))
BY
{ Auto }

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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. ∀v':V. (¬(COMMITED[v'] ∈ L))@i
7. (CONSIDERING[v] ∈ L)@i
8. filter(λx.cs-is-committed(x);L) = [] ∈ (consensus-state3(V) List)
9. filter(λx.cs-is-considering(x);L) = [] ∈ (consensus-state3(V) List)
10. ∀[i:ℕ||L||]. (¬↑cs-is-committed(L[i]))
11. ∀[i:ℕ||L||]. (¬↑cs-is-considering(L[i]))
12. True
13. True
⊢ AMBIVALENT = PREDECIDED[v] ∈ consensus-state2(V)

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Latex:

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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.  \mforall{}v:V.  ((COMMITED[v]  \mmember{}  L)  \mLeftarrow{}{}\mRightarrow{}  cs-ref-map3(L)  =  Decided[v])
6.  \mforall{}v':V.  (\mneg{}(COMMITED[v']  \mmember{}  L))@i
7.  (CONSIDERING[v]  \mmember{}  L)@i
8.  filter(\mlambda{}x.cs-is-committed(x);L)  =  []
9.  filter(\mlambda{}x.cs-is-considering(x);L)  =  []
\mvdash{}  uiff(True;\mforall{}[i:\mBbbN{}||L||].  (\mneg{}\muparrow{}cs-is-considering(L[i])))
{}\mRightarrow{}  uiff(True;\mforall{}[i:\mBbbN{}||L||].  (\mneg{}\muparrow{}cs-is-committed(L[i])))
{}\mRightarrow{}  (AMBIVALENT  =  PREDECIDED[v])


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Latex:
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