Proof of Nuprl Lemma : rsc4-validity-property

Step * 1 2 1 1 of Lemma rsc4-validity-property

1. Cmd : {T:Type| valueall-type(T)}
2. cmdeq : EqDecider(Cmd)
3. locs : bag(Id)
4. clients : bag(Id)
5. coeff : {2...}
6. flrs :

7. es : EO'
8. StandardAssumptions(rsc4_Main) es
9. e : E
10.

e1:E
((e1 < e)

((

v:

Cmd. (v

rsc4_decided'base(Cmd)(e1)

(

e':E. ((e' < e1)

v

rsc4_propose'base(Cmd)(e')))))

(

v:

Cmd

Id
(v

rsc4_vote'base(Cmd)(e1)

(

e':E. ((e' < e1)

<fst(fst(fst(v))), snd(fst(v))>

rsc4_propose'base(Cmd)(e')))))

(

v:

Cmd
(v

rsc4_retry'base(Cmd)(e1)

(

e':E. ((e' < e1)

<fst(fst(v)), snd(v)>

rsc4_propose'base(Cmd)(e')))))

(

e2:E
(e2 c

e1

(

cs:Cmd List.

L:Id List.

n,rnd:

.

c:Cmd.
(<cs, L>

rsc4_QuorumState(Cmd) <n, rnd>(e1)

(c

cs)

(

e':E. ((e' < e1)

<n, c>

rsc4_propose'base(Cmd)(e')))))))))
11. n :

12. c3 : Cmd
13. <n, c3>

rsc4_decided'base(Cmd)(e)
14. e' : E
15. (e' < e)
16. loc(e')

locs
17. loc(e) = loc(e')
18. e1 : {e1:E| e1

loc e' }
19. a :

20. a2 :

21. b2 :

List
22. <a2, b2>

Memory-class(rsc4_update_replica(Cmd);rsc4_init() <0, []>;rsc4_Proposal(Cmd))(e1)
23. (a2 < a)

(a

b2)
24. c2 : Cmd
25. True
26. <a, c2>

Base([propose];

Cmd)(e1)

(

b1:

.

b3:Id. <<<a, b1>, c2>, b3>

Base(``rsc4 vote``;

Cmd

Id)(e1))
27. e2 : {e1:E| e1

loc e' }
28. b3 :

29. (no rsc4_Quorum(Cmd;cmdeq;coeff;flrs) <a, b3> between e2 and e')
30. b5 :

31. c1 : Cmd
32. loc : Id
33. <<<n, b5>, c1>, loc>

Base(``rsc4 vote``;

Cmd

Id)(e')
34. cs : Cmd List
35. ||cs|| = (coeff * flrs)
36. (fst(poss-maj(cmdeq;[c1 / cs];c1))) = ((coeff * flrs) + 1)
37. c3 = (snd(poss-maj(cmdeq;[c1 / cs];c1)))
38. L : Id List
39. <cs, L>

rsc4_QuorumState(Cmd) <a, b3>(e')
40. a = n
41. b3 = b5
42.

(loc

L)
43. cmd : Cmd
44. True
45. <<a, b3>, cmd>

Base(``rsc4 retry``;

Cmd)(e2)

(

b5:Id. <<<a, b3>, cmd>, b5>

Base(``rsc4 vote``;

Cmd

Id)(e2))
46. b4 :

47. b4

Memory-class(rsc4_update_round(Cmd) a;rsc4_init() 0;rsc4_RoundInfo(Cmd))(e2)
48. b4 < b3
49. (no rsc4_Notify(Cmd;clients) a between e1 and e')
50. c3 = c1
51.

v:

Cmd. (v

rsc4_decided'base(Cmd)(e')

(

e'@0:E. ((e'@0 < e')

v

rsc4_propose'base(Cmd)(e'@0))))
52.

v:

Cmd

Id
(v

rsc4_vote'base(Cmd)(e')

(

e'@0:E. ((e'@0 < e')

<fst(fst(fst(v))), snd(fst(v))>

rsc4_propose'base(Cmd)(e'@0))))
53.

v:

Cmd
(v

rsc4_retry'base(Cmd)(e')

(

e'@0:E. ((e'@0 < e')

<fst(fst(v)), snd(v)>

rsc4_propose'base(Cmd)(e'@0))))
54.

e2:E
(e2 c

e'

(

cs:Cmd List.

L:Id List.

n,rnd:

.

c:Cmd.
(<cs, L>

rsc4_QuorumState(Cmd) <n, rnd>(e')

(c

cs)

(

e'@0:E. ((e'@0 < e')

<n, c>

rsc4_propose'base(Cmd)(e'@0))))))

e':E. ((e' < e)

<n, c3>

rsc4_propose'base(Cmd)(e'))
BY
{ (RepUR ``rsc4_vote'base`` -3
THEN OnMaybeHyp 33 (\h. ((FHyp (-3) [h] THENA Auto) THEN Reduce (-1) THEN RepeatFor 3 (ParallelLast) THEN MaAuto))
)

}

1. Cmd : \{T:Type| valueall-type(T)\} 2. cmdeq : EqDecider(Cmd) 3. locs : bag(Id) 4. clients : bag(Id) 5. coeff : \{2...\} 6. flrs : \mBbbN{} 7. es : EO' 8. StandardAssumptions(rsc4\_Main) es 9. e : E 10. \mforall{}e1:E ((e1 < e) {}\mRightarrow{} ((\mforall{}v:\mBbbZ{} \mtimes{} Cmd (v \mmember{} rsc4\_decided'base(Cmd)(e1) {}\mRightarrow{} (\mdownarrow{}\mexists{}e':E. ((e' < e1) \mwedge{} v \mmember{} rsc4\_propose'base(Cmd)(e'))))) \mwedge{} (\mforall{}v:\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id (v \mmember{} rsc4\_vote'base(Cmd)(e1) {}\mRightarrow{} (\mdownarrow{}\mexists{}e':E ((e' < e1) \mwedge{} <fst(fst(fst(v))), snd(fst(v))> \mmember{} rsc4\_propose'base(Cmd)(e'))))) \mwedge{} (\mforall{}v:\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd (v \mmember{} rsc4\_retry'base(Cmd)(e1) {}\mRightarrow{} (\mdownarrow{}\mexists{}e':E. ((e' < e1) \mwedge{} <fst(fst(v)), snd(v)> \mmember{} rsc4\_propose'base(Cmd)(e'))))) \mwedge{} (\mforall{}e2:E (e2 c\mleq{} e1 {}\mRightarrow{} (\mforall{}cs:Cmd List. \mforall{}L:Id List. \mforall{}n,rnd:\mBbbZ{}. \mforall{}c:Cmd. (<cs, L> \mmember{} rsc4\_QuorumState(Cmd) <n, rnd>(e1) {}\mRightarrow{} (c \mmember{} cs) {}\mRightarrow{} (\mdownarrow{}\mexists{}e':E. ((e' < e1) \mwedge{} <n, c> \mmember{} rsc4\_propose'base(Cmd)(e'))))))))) 11. n : \mBbbZ{} 12. c3 : Cmd 13. <n, c3> \mmember{} rsc4\_decided'base(Cmd)(e) 14. e' : E 15. (e' < e) 16. loc(e') \mdownarrow{}\mmember{} locs 17. loc(e) = loc(e') 18. e1 : \{e1:E| e1 \mleq{}loc e' \} 19. a : \mBbbZ{} 20. a2 : \mBbbZ{} 21. b2 : \mBbbZ{} List 22. <a2, b2> \mmember{} Memory-class(rsc4\_update\_replica(Cmd);rsc4\_init() ɘ, []>rsc4\_Proposal(Cmd))(e1) 23. (a2 < a) \mvee{} (a \mmember{} b2) 24. c2 : Cmd 25. True 26. <a, c2> \mmember{} Base([propose];\mBbbZ{} \mtimes{} Cmd)(e1) \mvee{} (\mexists{}b1:\mBbbZ{}. \mexists{}b3:Id. <<<a, b1>, c2>, b3> \mmember{} Base(``rsc4 vote``;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)(e1)) 27. e2 : \{e1:E| e1 \mleq{}loc e' \} 28. b3 : \mBbbZ{} 29. (no rsc4\_Quorum(Cmd;cmdeq;coeff;flrs) <a, b3> between e2 and e') 30. b5 : \mBbbZ{} 31. c1 : Cmd 32. loc : Id 33. <<<n, b5>, c1>, loc> \mmember{} Base(``rsc4 vote``;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)(e') 34. cs : Cmd List 35. ||cs|| = (coeff * flrs) 36. (fst(poss-maj(cmdeq;[c1 / cs];c1))) = ((coeff * flrs) + 1) 37. c3 = (snd(poss-maj(cmdeq;[c1 / cs];c1))) 38. L : Id List 39. <cs, L> \mmember{} rsc4\_QuorumState(Cmd) <a, b3>(e') 40. a = n 41. b3 = b5 42. \mneg{}(loc \mmember{} L) 43. cmd : Cmd 44. True 45. <<a, b3>, cmd> \mmember{} Base(``rsc4 retry``;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd)(e2) \mvee{} (\mexists{}b5:Id. <<<a, b3>, cmd>, b5> \mmember{} Base(``rsc4 vote``;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)(e2)) 46. b4 : \mBbbZ{} 47. b4 \mmember{} Memory-class(rsc4\_update\_round(Cmd) a;rsc4\_init() 0;rsc4\_RoundInfo(Cmd))(e2) 48. b4 < b3 49. (no rsc4\_Notify(Cmd;clients) a between e1 and e') 50. c3 = c1 51. \mforall{}v:\mBbbZ{} \mtimes{} Cmd (v \mmember{} rsc4\_decided'base(Cmd)(e') {}\mRightarrow{} (\mdownarrow{}\mexists{}e'@0:E. ((e'@0 < e') \mwedge{} v \mmember{} rsc4\_propose'base(Cmd)(e'@0)))) 52. \mforall{}v:\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id (v \mmember{} rsc4\_vote'base(Cmd)(e') {}\mRightarrow{} (\mdownarrow{}\mexists{}e'@0:E. ((e'@0 < e') \mwedge{} <fst(fst(fst(v))), snd(fst(v))> \mmember{} rsc4\_propose'base(Cmd)(e'@0)))) 53. \mforall{}v:\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd (v \mmember{} rsc4\_retry'base(Cmd)(e') {}\mRightarrow{} (\mdownarrow{}\mexists{}e'@0:E. ((e'@0 < e') \mwedge{} <fst(fst(v)), snd(v)> \mmember{} rsc4\_propose'base(Cmd)(e'@0)))) 54. \mforall{}e2:E (e2 c\mleq{} e' {}\mRightarrow{} (\mforall{}cs:Cmd List. \mforall{}L:Id List. \mforall{}n,rnd:\mBbbZ{}. \mforall{}c:Cmd. (<cs, L> \mmember{} rsc4\_QuorumState(Cmd) <n, rnd>(e') {}\mRightarrow{} (c \mmember{} cs) {}\mRightarrow{} (\mdownarrow{}\mexists{}e'@0:E. ((e'@0 < e') \mwedge{} <n, c> \mmember{} rsc4\_propose'base(Cmd)(e'@0)))))) \mvdash{} \mdownarrow{}\mexists{}e':E. ((e' < e) \mwedge{} <n, c3> \mmember{} rsc4\_propose'base(Cmd)(e')) By (RepUR ``rsc4\_vote'base`` -3 THEN OnMaybeHyp 33 (\mbackslash{}h. ((FHyp (-3) [h] THENA Auto) THEN Reduce (-1) THEN RepeatFor 3 (ParallelLast) THEN MaAuto)) )\mcdot{} Home Index