Step
*
2
1
1
2
1
3
of Lemma
rsc4-decided-property
1. Cmd : Type
2. Cmd 
ValueAllType
3. valueall-type(Cmd)
4. cmdeq : EqDecider(Cmd)
5. locs : bag(Id)
6. clients : bag(Id)
7. coeff : 
8. flrs :
9. es : EO'
10. StandardAssumptions(rsc4_Main) es
11. e : E
12. c : Cmd
13. n :
14. <n, c> rsc4_decided'base(Cmd)(e)
15. e' : E
16. (e' < e)
17. loc(e') 
locs
18. loc(e) = loc(e')
19. e1 : {e1:E| e1 
loc e' }
20. a :
21. a2 :
22. b2 : List
23. <a2, b2> Memory-class(rsc4_update_replica(Cmd);rsc4_init() <0, []>;rsc4_Proposal(Cmd))(e1)
24. (a2 < a) 
(a b2)
25. b : Cmd
26. True
27. <a, b> Base([propose]; 
Cmd)(e1)
(
b1:. b3:Id. <<<a, b1>, b>, b3> Base(``rsc4 vote``; Cmd Id)(e1))
28. b1 :
29. c1 : Cmd
30. lastloc : Id
31. <<<n, b1>, c1>, lastloc> Base(``rsc4 vote``; Cmd Id)(e')
32. cs : Cmd List
33. ||cs|| = (coeff * flrs)
34. (fst(poss-maj(cmdeq;[c1 / cs];c1))) = ((coeff * flrs) + 1)
35. c = (snd(poss-maj(cmdeq;[c1 / cs];c1)))
36. L : Id List
37. <cs, L> Memory-class(rsc4_add_to_quorum(Cmd) <a, 0>;rsc4_init() <[], []>;rsc4_vote'base(Cmd))(e')
38. a = n
39. 0 = b1
40. 
(lastloc L)
41. (no rsc4_Quorum(Cmd;cmdeq;coeff;flrs) <a, 0> between e1 and e')
42. c1 = c
43. 
z:Cmd. ((z cs) 
(z = c))
44. ||cs|| = ||L||
45. no_repeats(Id;L)
46. <cs, L> rsc4_QuorumState(Cmd) <a, 0>(e')
47. no_repeats(Id;L)
48. ||L|| = ||cs||
49. i:
||L||. (e:E. (e loc e' 
<<<a, 0>, cs[i]>, L[i]> rsc4_vote'base(Cmd)(e)))
50. vtr : Id
51. i :
52. i < ||L||
53. vtr = L[i]
54. e2 : E
55. e2 loc e'
56. <<<n, 0>, c>, vtr> rsc4_vote'base(Cmd)(e2)
57. cs[i] = c
58. e3 : E
59. (e3 < e2)
60. <loc(e2), info(e2)> rsc4_Main(e3)
e'@0:E. <loc(e'), make-Msg(``rsc4 vote``; Cmd Id;<<<n, 0>, c>, vtr>)> rsc4_Main(e'@0)
BY
{ (D 0 THEN (InstConcl [
e3
]
THENA MaAuto) THEN (Subst loc(e') = loc(e2) 0 THENA MaAuto)) }
1
1. Cmd : Type
2. Cmd ValueAllType
3. valueall-type(Cmd)
4. cmdeq : EqDecider(Cmd)
5. locs : bag(Id)
6. clients : bag(Id)
7. coeff :
8. flrs :
9. es : EO'
10. StandardAssumptions(rsc4_Main) es
11. e : E
12. c : Cmd
13. n :
14. <n, c> rsc4_decided'base(Cmd)(e)
15. e' : E
16. (e' < e)
17. loc(e') locs
18. loc(e) = loc(e')
19. e1 : {e1:E| e1 loc e' }
20. a :
21. a2 :
22. b2 : List
23. <a2, b2> Memory-class(rsc4_update_replica(Cmd);rsc4_init() <0, []>;rsc4_Proposal(Cmd))(e1)
24. (a2 < a) (a b2)
25. b : Cmd
26. True
27. <a, b> Base([propose]; Cmd)(e1)
(b1:. b3:Id. <<<a, b1>, b>, b3> Base(``rsc4 vote``; Cmd Id)(e1))
28. b1 :
29. c1 : Cmd
30. lastloc : Id
31. <<<n, b1>, c1>, lastloc> Base(``rsc4 vote``; Cmd Id)(e')
32. cs : Cmd List
33. ||cs|| = (coeff * flrs)
34. (fst(poss-maj(cmdeq;[c1 / cs];c1))) = ((coeff * flrs) + 1)
35. c = (snd(poss-maj(cmdeq;[c1 / cs];c1)))
36. L : Id List
37. <cs, L> Memory-class(rsc4_add_to_quorum(Cmd) <a, 0>;rsc4_init() <[], []>;rsc4_vote'base(Cmd))(e')
38. a = n
39. 0 = b1
40. (lastloc L)
41. (no rsc4_Quorum(Cmd;cmdeq;coeff;flrs) <a, 0> between e1 and e')
42. c1 = c
43. z:Cmd. ((z cs) (z = c))
44. ||cs|| = ||L||
45. no_repeats(Id;L)
46. <cs, L> rsc4_QuorumState(Cmd) <a, 0>(e')
47. no_repeats(Id;L)
48. ||L|| = ||cs||
49. i:||L||. (e:E. (e loc e' <<<a, 0>, cs[i]>, L[i]> rsc4_vote'base(Cmd)(e)))
50. vtr : Id
51. i :
52. i < ||L||
53. vtr = L[i]
54. e2 : E
55. e2 loc e'
56. <<<n, 0>, c>, vtr> rsc4_vote'base(Cmd)(e2)
57. cs[i] = c
58. e3 : E
59. (e3 < e2)
60. <loc(e2), info(e2)> rsc4_Main(e3)
<loc(e2), make-Msg(``rsc4 vote``; Cmd Id;<<<n, 0>, c>, vtr>)> rsc4_Main(e3)
1. Cmd : Type
2. Cmd \mmember{} ValueAllType
3. valueall-type(Cmd)
4. cmdeq : EqDecider(Cmd)
5. locs : bag(Id)
6. clients : bag(Id)
7. coeff : \mBbbZ{}
8. flrs : \mBbbZ{}
9. es : EO'
10. StandardAssumptions(rsc4\_Main) es
11. e : E
12. c : Cmd
13. n : \mBbbZ{}
14. <n, c> \mmember{} rsc4\_decided'base(Cmd)(e)
15. e' : E
16. (e' < e)
17. loc(e') \mdownarrow{}\mmember{} locs
18. loc(e) = loc(e')
19. e1 : \{e1:E| e1 \mleq{}loc e' \}
20. a : \mBbbZ{}
21. a2 : \mBbbZ{}
22. b2 : \mBbbZ{} List
23. <a2, b2> \mmember{} Memory-class(rsc4\_update\_replica(Cmd);rsc4\_init() ɘ, []>rsc4\_Proposal(Cmd))(e1)
24. (a2 < a) \mvee{} (a \mmember{} b2)
25. b : Cmd
26. True
27. <a, b> \mmember{} Base([propose];\mBbbZ{} \mtimes{} Cmd)(e1)
\mvee{} (\mexists{}b1:\mBbbZ{}. \mexists{}b3:Id. <<<a, b1>, b>, b3> \mmember{} Base(``rsc4 vote``;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)(e1))
28. b1 : \mBbbZ{}
29. c1 : Cmd
30. lastloc : Id
31. <<<n, b1>, c1>, lastloc> \mmember{} Base(``rsc4 vote``;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)(e')
32. cs : Cmd List
33. ||cs|| = (coeff * flrs)
34. (fst(poss-maj(cmdeq;[c1 / cs];c1))) = ((coeff * flrs) + 1)
35. c = (snd(poss-maj(cmdeq;[c1 / cs];c1)))
36. L : Id List
37. <cs, L> \mmember{} Memory-class(rsc4\_add\_to\_quorum(Cmd) <a, 0>rsc4\_init() <[], []>rsc4\_vote'base(Cmd))(
e')
38. a = n
39. 0 = b1
40. \mneg{}(lastloc \mmember{} L)
41. (no rsc4\_Quorum(Cmd;cmdeq;coeff;flrs) <a, 0> between e1 and e')
42. c1 = c
43. \mforall{}z:Cmd. ((z \mmember{} cs) {}\mRightarrow{} (z = c))
44. ||cs|| = ||L||
45. no\_repeats(Id;L)
46. <cs, L> \mmember{} rsc4\_QuorumState(Cmd) <a, 0>(e')
47. no\_repeats(Id;L)
48. ||L|| = ||cs||
49. \mforall{}i:\mBbbN{}||L||. (\mdownarrow{}\mexists{}e:E. (e \mleq{}loc e' \mwedge{} <<<a, 0>, cs[i]>, L[i]> \mmember{} rsc4\_vote'base(Cmd)(e)))
50. vtr : Id
51. i : \mBbbN{}
52. i < ||L||
53. vtr = L[i]
54. e2 : E
55. e2 \mleq{}loc e'
56. <<<n, 0>, c>, vtr> \mmember{} rsc4\_vote'base(Cmd)(e2)
57. cs[i] = c
58. e3 : E
59. (e3 < e2)
60. <loc(e2), info(e2)> \mmember{} rsc4\_Main(e3)
\mvdash{} \mdownarrow{}\mexists{}e'@0:E. <loc(e'), make-Msg(``rsc4 vote``;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id;<<<n, 0>, c>, vtr>)> \mmember{} rsc4\_Main(e'@0)
By
(D 0 THEN (InstConcl [\mkleeneopen{}e3\mkleeneclose{}]\mcdot{} THENA MaAuto) THEN (Subst \mkleeneopen{}loc(e') = loc(e2)\mkleeneclose{} 0\mcdot{} THENA MaAuto))
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