Proof of Nuprl Lemma : cs-ref-map-unchanged

Step * 3 of Lemma cs-ref-map-unchanged

1. [V] : Type
2. ∀v1,v2:V.  Dec(v1 = v2 ∈ V)@i
3. A : Id List@i
4. W : {a:Id| (a ∈ A)}  List List@i
5. two-intersection(A;W)@i
6. f : ConsensusState ─→ (consensus-state3(V) List)@i
7. ∀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          ∧ (f s[i] = WITHDRAWN ∈ consensus-state3(V) ⇐⇒ ¬(∃v:V. in state s, inning i could commit v ))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
8. x : ts-reachable(consensus-ts4(V;A;W))@i
9. y : ts-reachable(consensus-ts4(V;A;W))@i
10. x ∈ ConsensusState
11. y ∈ ConsensusState
12. x ts-rel(consensus-ts4(V;A;W)) y@i
13. i : ℕ@i
14. i < ||f x||
15. i < ||f y||
16. ∀v:V. (in state x, inning i could commit v  ⇒ in state y, inning i could commit v )@i
17. v : V
18. f x[i] = CONSIDERING[v] ∈ consensus-state3(V)
19. (f x[i] = INITIAL ∈ consensus-state3(V))
⇒

 (∃v,v':V. ((¬(v = v' ∈ V)) ∧ in state x, inning i could commit v  ∧ in state x, inning i could commit v' ))

20. (f x[i] = INITIAL ∈ consensus-state3(V)) ⇐ ∃v,v':V

                                                 ((¬(v = v' ∈ V))

                                                 ∧ in state x, inning i could commit v

                                                 ∧ in state x, inning i could commit 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 (f x[i] = WITHDRAWN ∈ consensus-state3(V)) ⇒ (¬(∃v:V. in state x, inning i could commit v ))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 (f x[i] = WITHDRAWN ∈ consensus-state3(V)) ⇐ ¬(∃v:V. in state x, inning i could commit v )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. (f x[i] = COMMITED[v] ∈ consensus-state3(V)) ⇒ in state x, inning i has committed v
25. (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)))
26. in state x, inning i could commit v
27. ¬in state x, inning i has committed v
28. ∀v':V. (in state x, inning i could commit v'  ⇒ (v' = v ∈ V))
⊢ (f y[i] = f x[i] ∈ consensus-state3(V))

∨ (∃v:V. ((f y[i] = COMMITED[v] ∈ consensus-state3(V)) ∧ (f x[i] = CONSIDERING[v] ∈ consensus-state3(V))))

BY
{ (Decide in state y, inning i has committed v THENA Auto) }

1
1. [V] : Type
2. ∀v1,v2:V.  Dec(v1 = v2 ∈ V)@i
3. A : Id List@i
4. W : {a:Id| (a ∈ A)}  List List@i
5. two-intersection(A;W)@i
6. f : ConsensusState ─→ (consensus-state3(V) List)@i
7. ∀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' ))

 (∃v,v':V. ((¬(v = v' ∈ V)) ∧ in state x, inning i could commit v  ∧ in state x, inning i could commit v' ))

20. (f x[i] = INITIAL ∈ consensus-state3(V)) ⇐ ∃v,v':V

                                                 ((¬(v = v' ∈ V))

                                                 ∧ in state x, inning i could commit v

                                                 ∧ in state x, inning i could commit v' )

∨ (∃v:V. ((f y[i] = COMMITED[v] ∈ consensus-state3(V)) ∧ (f x[i] = CONSIDERING[v] ∈ consensus-state3(V))))

2
1. [V] : Type
2. ∀v1,v2:V.  Dec(v1 = v2 ∈ V)@i
3. A : Id List@i
4. W : {a:Id| (a ∈ A)}  List List@i
5. two-intersection(A;W)@i
6. f : ConsensusState ─→ (consensus-state3(V) List)@i
7. ∀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' ))

 (∃v,v':V. ((¬(v = v' ∈ V)) ∧ in state x, inning i could commit v  ∧ in state x, inning i could commit v' ))

20. (f x[i] = INITIAL ∈ consensus-state3(V)) ⇐ ∃v,v':V

                                                 ((¬(v = v' ∈ V))

                                                 ∧ in state x, inning i could commit v

                                                 ∧ in state x, inning i could commit v' )

∨ (∃v:V. ((f y[i] = COMMITED[v] ∈ consensus-state3(V)) ∧ (f x[i] = CONSIDERING[v] ∈ consensus-state3(V))))

Latex:

1. [V] : Type 2. \mforall{}v1,v2:V. Dec(v1 = v2)@i 3. A : Id List@i 4. W : \{a:Id| (a \mmember{} A)\} List List@i 5. two-intersection(A;W)@i 6. f : ConsensusState {}\mrightarrow{} (consensus-state3(V) List)@i 7. \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 8. x : ts-reachable(consensus-ts4(V;A;W))@i 9. y : ts-reachable(consensus-ts4(V;A;W))@i 10. x \mmember{} ConsensusState 11. y \mmember{} ConsensusState 12. x ts-rel(consensus-ts4(V;A;W)) y@i 13. i : \mBbbN{}@i 14. i < ||f x|| 15. i < ||f y|| 16. \mforall{}v:V. (in state x, inning i could commit v {}\mRightarrow{} in state y, inning i could commit v )@i 17. v : V 18. f x[i] = CONSIDERING[v] 19. (f x[i] = INITIAL) {}\mRightarrow{} (\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' )) 20. (f x[i] = INITIAL) \mLeftarrow{}{} \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' ) 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. (f x[i] = COMMITED[v]) {}\mRightarrow{} in state x, inning i has committed v 25. (f x[i] = CONSIDERING[v]) \mLeftarrow{}{} 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))) 26. in state x, inning i could commit v 27. \mneg{}in state x, inning i has committed v 28. \mforall{}v':V. (in state x, inning i could commit v' {}\mRightarrow{} (v' = v)) \mvdash{} (f y[i] = f x[i]) \mvee{} (\mexists{}v:V. ((f y[i] = COMMITED[v]) \mwedge{} (f x[i] = CONSIDERING[v]))) By (Decide in state y, inning i has committed v THENA Auto) Home Index