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Step * 1 1 6 1 1 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. False supposing ∀[i:ℕ||L||]. (¬↑cs-is-committed(L[i]))@i
7. cs-committed-val(hd(filter(λx.cs-is-committed(x);L))) = v ∈ V
⊢ (COMMITED[cs-committed-val(hd(filter(λx.cs-is-committed(x);L)))] ∈ L)
BY
{ (MoveToConcl (-3)
   THEN (InstLemma `filter_type` [⌈consensus-state3(V)⌉;⌈λx.cs-is-committed(x)⌉;⌈L⌉]⋅ THENA Auto)
   THEN (GenConclAtAddr [1;1;2] THENA (Reduce 0 THEN Auto))
   THEN D -2
   THEN Reduce 0
   THEN Auto) }

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. False supposing ∀[i:ℕ||L||]. (¬↑cs-is-committed(L[i]))@i
6. cs-committed-val(hd(filter(λx.cs-is-committed(x);L))) = v ∈ V
7. filter(λx.cs-is-committed(x);L) ∈ {x:consensus-state3(V)| ↑((λx.cs-is-committed(x)) x)}  List
8. u : {x:consensus-state3(V)| ↑((λx.cs-is-committed(x)) x)} 
9. v2 : {x:consensus-state3(V)| ↑((λx.cs-is-committed(x)) x)}  List
10. filter(λx.cs-is-committed(x);L) = [u / v2] ∈ ({x:consensus-state3(V)| ↑((λx.cs-is-committed(x)) x)}  List)@i
11. ¬([u / v2] = [] ∈ (consensus-state3(V) List))@i
⊢ (COMMITED[cs-committed-val(u)] ∈ L)

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


By

(MoveToConcl  (-3)
  THEN  (InstLemma  `filter\_type`  [\mkleeneopen{}consensus-state3(V)\mkleeneclose{};\mkleeneopen{}\mlambda{}x.cs-is-committed(x)\mkleeneclose{};\mkleeneopen{}L\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (GenConclAtAddr  [1;1;2]  THENA  (Reduce  0  THEN  Auto))
  THEN  D  -2
  THEN  Reduce  0
  THEN  Auto)

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