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Step * 1 2 2 1 2 1 1 1 of Lemma consensus-ts3-invariant1


1. V : Type@i'
2. L : consensus-state3(V) List@i
3. y : consensus-state3(V) List@i
4. ∀v:V
     (((CONSIDERING[v] ∈ L) ∨ (COMMITED[v] ∈ L))
     ⇒ (∀v':V. (((CONSIDERING[v'] ∈ L) ∨ (COMMITED[v'] ∈ L)) ⇒ (v' = v ∈ V))))@i
5. ||y|| = ||L|| ∈ ℤ@i
6. i : ℕ||L||@i
7. ∀j:ℕ||L||. ((¬(j = i ∈ ℤ)) ⇒ (y[j] = L[j] ∈ consensus-state3(V)))@i
8. L[i] = INITIAL ∈ consensus-state3(V)@i
9. v1 : V@i
10. y[i] = CONSIDERING[v1] ∈ consensus-state3(V)@i
11. ∀j:ℕi
      ((L[j] = WITHDRAWN ∈ consensus-state3(V))
      ∨ (L[j] = CONSIDERING[v1] ∈ consensus-state3(V))
      ∨ (L[j] = COMMITED[v1] ∈ consensus-state3(V)))@i
12. ∀j:ℕ||L||. (L[j] = INITIAL ∈ consensus-state3(V)) ∨ (L[j] = WITHDRAWN ∈ consensus-state3(V)) supposing i < j
13. j : ℕ||L||@i
14. v : V@i
15. (y[j] = CONSIDERING[v] ∈ consensus-state3(V)) ∨ (y[j] = COMMITED[v] ∈ consensus-state3(V))@i
⊢ v = v1 ∈ V
BY
{ ((InstLemma `consensus-state3-unequal` [⌈V⌉]⋅ THENA Auto)
   THEN D -1
   THEN InstHyp
   [⌈v⌉] (-1)⋅
   THEN Auto
   THEN (InstHyp [⌈v1⌉] (-1)⋅ THENA Auto)
   THEN D -1) }

1
1. V : Type@i'
2. L : consensus-state3(V) List@i
3. y : consensus-state3(V) List@i
4. ∀v:V
     (((CONSIDERING[v] ∈ L) ∨ (COMMITED[v] ∈ L))
     ⇒ (∀v':V. (((CONSIDERING[v'] ∈ L) ∨ (COMMITED[v'] ∈ L)) ⇒ (v' = v ∈ V))))@i
5. ||y|| = ||L|| ∈ ℤ@i
6. i : ℕ||L||@i
7. ∀j:ℕ||L||. ((¬(j = i ∈ ℤ)) ⇒ (y[j] = L[j] ∈ consensus-state3(V)))@i
8. L[i] = INITIAL ∈ consensus-state3(V)@i
9. v1 : V@i
10. y[i] = CONSIDERING[v1] ∈ consensus-state3(V)@i
11. ∀j:ℕi
      ((L[j] = WITHDRAWN ∈ consensus-state3(V))
      ∨ (L[j] = CONSIDERING[v1] ∈ consensus-state3(V))
      ∨ (L[j] = COMMITED[v1] ∈ consensus-state3(V)))@i
12. ∀j:ℕ||L||. (L[j] = INITIAL ∈ consensus-state3(V)) ∨ (L[j] = WITHDRAWN ∈ consensus-state3(V)) supposing i < j
13. j : ℕ||L||@i
14. v : V@i
15. (y[j] = CONSIDERING[v] ∈ consensus-state3(V)) ∨ (y[j] = COMMITED[v] ∈ consensus-state3(V))@i
16. ¬(INITIAL = WITHDRAWN ∈ consensus-state3(V))
17. ∀[v:V]
      (((¬(COMMITED[v] = INITIAL ∈ consensus-state3(V))) ∧ (¬(CONSIDERING[v] = INITIAL ∈ consensus-state3(V))))
      ∧ (¬(COMMITED[v] = WITHDRAWN ∈ consensus-state3(V)))
      ∧ (¬(CONSIDERING[v] = WITHDRAWN ∈ consensus-state3(V)))
      ∧ (∀[v':V]
           ((¬(CONSIDERING[v] = COMMITED[v'] ∈ consensus-state3(V)))
           ∧ (¬(CONSIDERING[v] = CONSIDERING[v'] ∈ consensus-state3(V)))
             ∧ (¬(COMMITED[v] = COMMITED[v'] ∈ consensus-state3(V))) 
             supposing ¬(v = v' ∈ V))))
18. ¬(COMMITED[v] = INITIAL ∈ consensus-state3(V))
19. ¬(CONSIDERING[v] = INITIAL ∈ consensus-state3(V))
20. ¬(COMMITED[v] = WITHDRAWN ∈ consensus-state3(V))
21. ¬(CONSIDERING[v] = WITHDRAWN ∈ consensus-state3(V))
22. ∀[v':V]
      ((¬(CONSIDERING[v] = COMMITED[v'] ∈ consensus-state3(V)))
      ∧ (¬(CONSIDERING[v] = CONSIDERING[v'] ∈ consensus-state3(V)))
        ∧ (¬(COMMITED[v] = COMMITED[v'] ∈ consensus-state3(V))) 
        supposing ¬(v = v' ∈ V))
23. ¬(CONSIDERING[v] = COMMITED[v1] ∈ consensus-state3(V))
24. (¬(CONSIDERING[v] = CONSIDERING[v1] ∈ consensus-state3(V))) ∧ (¬(COMMITED[v] = COMMITED[v1] ∈ consensus-state3(V))) 
    supposing ¬(v = v1 ∈ V)
⊢ v = v1 ∈ V

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Latex:



1.  V  :  Type@i'
2.  L  :  consensus-state3(V)  List@i
3.  y  :  consensus-state3(V)  List@i
4.  \mforall{}v:V
          (((CONSIDERING[v]  \mmember{}  L)  \mvee{}  (COMMITED[v]  \mmember{}  L))
          {}\mRightarrow{}  (\mforall{}v':V.  (((CONSIDERING[v']  \mmember{}  L)  \mvee{}  (COMMITED[v']  \mmember{}  L))  {}\mRightarrow{}  (v'  =  v))))@i
5.  ||y||  =  ||L||@i
6.  i  :  \mBbbN{}||L||@i
7.  \mforall{}j:\mBbbN{}||L||.  ((\mneg{}(j  =  i))  {}\mRightarrow{}  (y[j]  =  L[j]))@i
8.  L[i]  =  INITIAL@i
9.  v1  :  V@i
10.  y[i]  =  CONSIDERING[v1]@i
11.  \mforall{}j:\mBbbN{}i.  ((L[j]  =  WITHDRAWN)  \mvee{}  (L[j]  =  CONSIDERING[v1])  \mvee{}  (L[j]  =  COMMITED[v1]))@i
12.  \mforall{}j:\mBbbN{}||L||.  (L[j]  =  INITIAL)  \mvee{}  (L[j]  =  WITHDRAWN)  supposing  i  <  j
13.  j  :  \mBbbN{}||L||@i
14.  v  :  V@i
15.  (y[j]  =  CONSIDERING[v])  \mvee{}  (y[j]  =  COMMITED[v])@i
\mvdash{}  v  =  v1


By

((InstLemma  `consensus-state3-unequal`  [\mkleeneopen{}V\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  D  -1
  THEN  InstHyp
  [\mkleeneopen{}v\mkleeneclose{}]  (-1)\mcdot{}
  THEN  Auto
  THEN  (InstHyp  [\mkleeneopen{}v1\mkleeneclose{}]  (-1)\mcdot{}  THENA  Auto)
  THEN  D  -1)

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