Step
*
1
1
2
2
2
of Lemma
cs-ref-map-changed
1. [V] : Type
2. ∀v1,v2:V. Dec(v1 = v2 ∈ V)@i
3. {∃v,v':V. (¬(v = v' ∈ V))}@i
4. A : Id List@i
5. W : {a:Id| (a ∈ A)} List List@i
6. two-intersection(A;W)@i
7. f : ConsensusState ─→ (consensus-state3(V) List)@i
8. ∀s:ConsensusState. ∀i:ℕ.
((i < ||f s|| ⇐⇒ ∃a:{a:Id| (a ∈ A)} . (i ≤ Inning(s;a)))
∧ (i < ||f s||
⇒ ((f s[i] = INITIAL ∈ consensus-state3(V)
⇐⇒ ∃v,v':V
((¬(v = v' ∈ V)) ∧ in state s, inning i could commit v ∧ in state s, inning i could commit v' ))
∧ (f s[i] = WITHDRAWN ∈ consensus-state3(V) ⇐⇒ ¬(∃v:V. in state s, inning i could commit v ))
∧ (∀v:V
((f s[i] = COMMITED[v] ∈ consensus-state3(V) ⇐⇒ in state s, inning i has committed v)
∧ (f s[i] = CONSIDERING[v] ∈ consensus-state3(V)
⇐⇒ in state s, inning i could commit v
∧ (¬in state s, inning i has committed v)
∧ (∀v':V. (in state s, inning i could commit v' ⇒ (v' = v ∈ V)))))))))@i
9. x : ts-reachable(consensus-ts4(V;A;W))@i
10. y : ts-reachable(consensus-ts4(V;A;W))@i
11. x ∈ ConsensusState
12. y ∈ ConsensusState
13. x ts-rel(consensus-ts4(V;A;W)) y@i
14. i : ℕ@i
15. i < ||f x||
16. i < ||f y||
17. v : V@i
18. in state x, inning i could commit v @i
19. ¬in state y, inning i could commit v @i
20. f x[i] = INITIAL ∈ consensus-state3(V)
21. (f x[i] = WITHDRAWN ∈ consensus-state3(V)) ⇒ (¬(∃v:V. in state x, inning i could commit v ))
22. (f x[i] = WITHDRAWN ∈ consensus-state3(V)) ⇐ ¬(∃v:V. in state x, inning i could commit v )
23. ∀v:V
((f x[i] = COMMITED[v] ∈ consensus-state3(V) ⇐⇒ in state x, inning i has committed v)
∧ (f x[i] = CONSIDERING[v] ∈ consensus-state3(V)
⇐⇒ in state x, inning i could commit v
∧ (¬in state x, inning i has committed v)
∧ (∀v':V. (in state x, inning i could commit v' ⇒ (v' = v ∈ V)))))
24. ∃v,v':V. ((¬(v = v' ∈ V)) ∧ in state x, inning i could commit v ∧ in state x, inning i could commit v' )
25. f x[i] = INITIAL ∈ consensus-state3(V)
26. ¬(∃v,v':V. ((¬(v = v' ∈ V)) ∧ in state y, inning i could commit v ∧ in state y, inning i could commit v' ))
27. Dec(∃v:V. in state y, inning i could commit v )
⊢ (f y[i] = WITHDRAWN ∈ consensus-state3(V))
∨ ((f x[i] = INITIAL ∈ consensus-state3(V))
∧ ((f y[i] = INITIAL ∈ consensus-state3(V))
∨ (∃v':V
((∀j:ℕi. (¬(f x[j] = INITIAL ∈ consensus-state3(V))))
∧ ((f y[i] = CONSIDERING[v'] ∈ consensus-state3(V)) ∨ (f y[i] = COMMITED[v'] ∈ consensus-state3(V)))
∧ (∀j:ℕi. ∀v'':V.
(((f x[j] = CONSIDERING[v''] ∈ consensus-state3(V)) ∨ (f x[j] = COMMITED[v''] ∈ consensus-state3(V)))
⇒ (v'' = v' ∈ V)))))))
BY
{ D -1 }
1
1. [V] : Type
2. ∀v1,v2:V. Dec(v1 = v2 ∈ V)@i
3. {∃v,v':V. (¬(v = v' ∈ V))}@i
4. A : Id List@i
5. W : {a:Id| (a ∈ A)} List List@i
6. two-intersection(A;W)@i
7. f : ConsensusState ─→ (consensus-state3(V) List)@i
8. ∀s:ConsensusState. ∀i:ℕ.
((i < ||f s|| ⇐⇒ ∃a:{a:Id| (a ∈ A)} . (i ≤ Inning(s;a)))
∧ (i < ||f s||
⇒ ((f s[i] = INITIAL ∈ consensus-state3(V)
⇐⇒ ∃v,v':V
((¬(v = v' ∈ V)) ∧ in state s, inning i could commit v ∧ in state s, inning i could commit v' ))
∧ (f s[i] = WITHDRAWN ∈ consensus-state3(V) ⇐⇒ ¬(∃v:V. in state s, inning i could commit v ))
∧ (∀v:V
((f s[i] = COMMITED[v] ∈ consensus-state3(V) ⇐⇒ in state s, inning i has committed v)
∧ (f s[i] = CONSIDERING[v] ∈ consensus-state3(V)
⇐⇒ in state s, inning i could commit v
∧ (¬in state s, inning i has committed v)
∧ (∀v':V. (in state s, inning i could commit v' ⇒ (v' = v ∈ V)))))))))@i
9. x : ts-reachable(consensus-ts4(V;A;W))@i
10. y : ts-reachable(consensus-ts4(V;A;W))@i
11. x ∈ ConsensusState
12. y ∈ ConsensusState
13. x ts-rel(consensus-ts4(V;A;W)) y@i
14. i : ℕ@i
15. i < ||f x||
16. i < ||f y||
17. v : V@i
18. in state x, inning i could commit v @i
19. ¬in state y, inning i could commit v @i
20. f x[i] = INITIAL ∈ consensus-state3(V)
21. (f x[i] = WITHDRAWN ∈ consensus-state3(V)) ⇒ (¬(∃v:V. in state x, inning i could commit v ))
22. (f x[i] = WITHDRAWN ∈ consensus-state3(V)) ⇐ ¬(∃v:V. in state x, inning i could commit v )
23. ∀v:V
((f x[i] = COMMITED[v] ∈ consensus-state3(V) ⇐⇒ in state x, inning i has committed v)
∧ (f x[i] = CONSIDERING[v] ∈ consensus-state3(V)
⇐⇒ in state x, inning i could commit v
∧ (¬in state x, inning i has committed v)
∧ (∀v':V. (in state x, inning i could commit v' ⇒ (v' = v ∈ V)))))
24. ∃v,v':V. ((¬(v = v' ∈ V)) ∧ in state x, inning i could commit v ∧ in state x, inning i could commit v' )
25. f x[i] = INITIAL ∈ consensus-state3(V)
26. ¬(∃v,v':V. ((¬(v = v' ∈ V)) ∧ in state y, inning i could commit v ∧ in state y, inning i could commit v' ))
27. ∃v:V. in state y, inning i could commit v
⊢ (f y[i] = WITHDRAWN ∈ consensus-state3(V))
∨ ((f x[i] = INITIAL ∈ consensus-state3(V))
∧ ((f y[i] = INITIAL ∈ consensus-state3(V))
∨ (∃v':V
((∀j:ℕi. (¬(f x[j] = INITIAL ∈ consensus-state3(V))))
∧ ((f y[i] = CONSIDERING[v'] ∈ consensus-state3(V)) ∨ (f y[i] = COMMITED[v'] ∈ consensus-state3(V)))
∧ (∀j:ℕi. ∀v'':V.
(((f x[j] = CONSIDERING[v''] ∈ consensus-state3(V)) ∨ (f x[j] = COMMITED[v''] ∈ consensus-state3(V)))
⇒ (v'' = v' ∈ V)))))))
2
1. [V] : Type
2. ∀v1,v2:V. Dec(v1 = v2 ∈ V)@i
3. {∃v,v':V. (¬(v = v' ∈ V))}@i
4. A : Id List@i
5. W : {a:Id| (a ∈ A)} List List@i
6. two-intersection(A;W)@i
7. f : ConsensusState ─→ (consensus-state3(V) List)@i
8. ∀s:ConsensusState. ∀i:ℕ.
((i < ||f s|| ⇐⇒ ∃a:{a:Id| (a ∈ A)} . (i ≤ Inning(s;a)))
∧ (i < ||f s||
⇒ ((f s[i] = INITIAL ∈ consensus-state3(V)
⇐⇒ ∃v,v':V
((¬(v = v' ∈ V)) ∧ in state s, inning i could commit v ∧ in state s, inning i could commit v' ))
∧ (f s[i] = WITHDRAWN ∈ consensus-state3(V) ⇐⇒ ¬(∃v:V. in state s, inning i could commit v ))
∧ (∀v:V
((f s[i] = COMMITED[v] ∈ consensus-state3(V) ⇐⇒ in state s, inning i has committed v)
∧ (f s[i] = CONSIDERING[v] ∈ consensus-state3(V)
⇐⇒ in state s, inning i could commit v
∧ (¬in state s, inning i has committed v)
∧ (∀v':V. (in state s, inning i could commit v' ⇒ (v' = v ∈ V)))))))))@i
9. x : ts-reachable(consensus-ts4(V;A;W))@i
10. y : ts-reachable(consensus-ts4(V;A;W))@i
11. x ∈ ConsensusState
12. y ∈ ConsensusState
13. x ts-rel(consensus-ts4(V;A;W)) y@i
14. i : ℕ@i
15. i < ||f x||
16. i < ||f y||
17. v : V@i
18. in state x, inning i could commit v @i
19. ¬in state y, inning i could commit v @i
20. f x[i] = INITIAL ∈ consensus-state3(V)
21. (f x[i] = WITHDRAWN ∈ consensus-state3(V)) ⇒ (¬(∃v:V. in state x, inning i could commit v ))
22. (f x[i] = WITHDRAWN ∈ consensus-state3(V)) ⇐ ¬(∃v:V. in state x, inning i could commit v )
23. ∀v:V
((f x[i] = COMMITED[v] ∈ consensus-state3(V) ⇐⇒ in state x, inning i has committed v)
∧ (f x[i] = CONSIDERING[v] ∈ consensus-state3(V)
⇐⇒ in state x, inning i could commit v
∧ (¬in state x, inning i has committed v)
∧ (∀v':V. (in state x, inning i could commit v' ⇒ (v' = v ∈ V)))))
24. ∃v,v':V. ((¬(v = v' ∈ V)) ∧ in state x, inning i could commit v ∧ in state x, inning i could commit v' )
25. f x[i] = INITIAL ∈ consensus-state3(V)
26. ¬(∃v,v':V. ((¬(v = v' ∈ V)) ∧ in state y, inning i could commit v ∧ in state y, inning i could commit v' ))
27. ¬(∃v:V. in state y, inning i could commit v )
⊢ (f y[i] = WITHDRAWN ∈ consensus-state3(V))
∨ ((f x[i] = INITIAL ∈ consensus-state3(V))
∧ ((f y[i] = INITIAL ∈ consensus-state3(V))
∨ (∃v':V
((∀j:ℕi. (¬(f x[j] = INITIAL ∈ consensus-state3(V))))
∧ ((f y[i] = CONSIDERING[v'] ∈ consensus-state3(V)) ∨ (f y[i] = COMMITED[v'] ∈ consensus-state3(V)))
∧ (∀j:ℕi. ∀v'':V.
(((f x[j] = CONSIDERING[v''] ∈ consensus-state3(V)) ∨ (f x[j] = COMMITED[v''] ∈ consensus-state3(V)))
⇒ (v'' = v' ∈ V)))))))
Latex:
1. [V] : Type
2. \mforall{}v1,v2:V. Dec(v1 = v2)@i
3. \{\mexists{}v,v':V. (\mneg{}(v = v'))\}@i
4. A : Id List@i
5. W : \{a:Id| (a \mmember{} A)\} List List@i
6. two-intersection(A;W)@i
7. f : ConsensusState {}\mrightarrow{} (consensus-state3(V) List)@i
8. \mforall{}s:ConsensusState. \mforall{}i:\mBbbN{}.
((i < ||f s|| \mLeftarrow{}{}\mRightarrow{} \mexists{}a:\{a:Id| (a \mmember{} A)\} . (i \mleq{} Inning(s;a)))
\mwedge{} (i < ||f s||
{}\mRightarrow{} ((f s[i] = INITIAL
\mLeftarrow{}{}\mRightarrow{} \mexists{}v,v':V
((\mneg{}(v = v'))
\mwedge{} in state s, inning i could commit v
\mwedge{} in state s, inning i could commit v' ))
\mwedge{} (f s[i] = WITHDRAWN \mLeftarrow{}{}\mRightarrow{} \mneg{}(\mexists{}v:V. in state s, inning i could commit v ))
\mwedge{} (\mforall{}v:V
((f s[i] = COMMITED[v] \mLeftarrow{}{}\mRightarrow{} in state s, inning i has committed v)
\mwedge{} (f s[i] = CONSIDERING[v]
\mLeftarrow{}{}\mRightarrow{} in state s, inning i could commit v
\mwedge{} (\mneg{}in state s, inning i has committed v)
\mwedge{} (\mforall{}v':V. (in state s, inning i could commit v' {}\mRightarrow{} (v' = v)))))))))@i
9. x : ts-reachable(consensus-ts4(V;A;W))@i
10. y : ts-reachable(consensus-ts4(V;A;W))@i
11. x \mmember{} ConsensusState
12. y \mmember{} ConsensusState
13. x ts-rel(consensus-ts4(V;A;W)) y@i
14. i : \mBbbN{}@i
15. i < ||f x||
16. i < ||f y||
17. v : V@i
18. in state x, inning i could commit v @i
19. \mneg{}in state y, inning i could commit v @i
20. f x[i] = INITIAL
21. (f x[i] = WITHDRAWN) {}\mRightarrow{} (\mneg{}(\mexists{}v:V. in state x, inning i could commit v ))
22. (f x[i] = WITHDRAWN) \mLeftarrow{}{} \mneg{}(\mexists{}v:V. in state x, inning i could commit v )
23. \mforall{}v:V
((f x[i] = COMMITED[v] \mLeftarrow{}{}\mRightarrow{} in state x, inning i has committed v)
\mwedge{} (f x[i] = CONSIDERING[v]
\mLeftarrow{}{}\mRightarrow{} in state x, inning i could commit v
\mwedge{} (\mneg{}in state x, inning i has committed v)
\mwedge{} (\mforall{}v':V. (in state x, inning i could commit v' {}\mRightarrow{} (v' = v)))))
24. \mexists{}v,v':V
((\mneg{}(v = v')) \mwedge{} in state x, inning i could commit v \mwedge{} in state x, inning i could commit v' )
25. f x[i] = INITIAL
26. \mneg{}(\mexists{}v,v':V
((\mneg{}(v = v')) \mwedge{} in state y, inning i could commit v \mwedge{} in state y, inning i could commit v' ))
27. Dec(\mexists{}v:V. in state y, inning i could commit v )
\mvdash{} (f y[i] = WITHDRAWN)
\mvee{} ((f x[i] = INITIAL)
\mwedge{} ((f y[i] = INITIAL)
\mvee{} (\mexists{}v':V
((\mforall{}j:\mBbbN{}i. (\mneg{}(f x[j] = INITIAL)))
\mwedge{} ((f y[i] = CONSIDERING[v']) \mvee{} (f y[i] = COMMITED[v']))
\mwedge{} (\mforall{}j:\mBbbN{}i. \mforall{}v'':V.
(((f x[j] = CONSIDERING[v'']) \mvee{} (f x[j] = COMMITED[v''])) {}\mRightarrow{} (v'' = v')))))))
By
D -1
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