Step
*
1
of Lemma
new_23_sig_retry_property
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : ℕ
6. flrs : ℕ
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig{i:l}(ℤ)
10. f : new_23_sig_headers_type{i:l}(Cmd;notify;propose)
11. es : EO+(Message(f))
12. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. e : E@i
14. c : Cmd@i
15. n : ℤ@i
16. r : ℤ@i
17. <<n, r + 1>, c> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)@i
18. (f propose) = (ℤ × Cmd) ∈ Type
19. (f notify) = (ℤ × Cmd) ∈ Type
20. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
21. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
22. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
23. f ∈ Name ─→ Type
24. e' : E
25. delay : ℤ
26. (e' < e)
27. loc(e') ↓∈ reps
28. e1 : {e1:E| e1 ≤loc e' }
29. z1 : ℤ
30. z2 : Cmd
31. ((header(e1) = propose ∈ Name) ∧ has-es-info-type(es;e1;f;ℤ × Cmd) ∧ (<z1, z2> = msgval(e1) ∈ (ℤ × Cmd)))
∨ ((header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (z1 = (fst(fst(fst(msgval(e1))))) ∈ ℤ)
∧ (z2 = (snd(fst(msgval(e1)))) ∈ Cmd))
32. ¬↑(set-sig-member(slots) z1 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
33. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
34. (no new_23_sig_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, 0> between e1 and e')
35. header(e') = ``new_23_sig vote`` ∈ Name
36. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
37. <z1, 0> = (fst(fst(msgval(e')))) ∈ (ℤ × ℤ)
38. ¬(snd(msgval(e')) ∈ snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))
39. ||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = (coeff * flrs) ∈ ℤ
40. delay = 0 ∈ ℤ
41. ¬((fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))];\000Csnd(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ)
42. loc(e) = loc(e') ∈ Id
43. msg-authentic(info(e)) = ff
44. n = (fst(fst(fst(msgval(e'))))) ∈ ℤ
45. (r + 1) = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ
46. c = (snd(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))\000C];snd(fst(msgval(e')))))) ∈ Cmd
47. 0 ≤ (coeff * flrs)
⊢ ↓∃x:Id
∃cs:Cmd List+
∃L:Id List
((no_repeats(Id;L) ∧ (||L|| = ((coeff * flrs) + 1) ∈ ℤ) ∧ (||cs|| = ||L|| ∈ ℤ))
∧ (∀i:ℕ||L||. ((↓∃e':E. mk-msg-interface(x;make-Msg(``new_23_sig vote``;<<<n, r>, cs[i]>, L[i]>)) ∈ new_23_sig_mai\000Cn()(e')) ∧ L[i] ↓∈ reps))
∧ (c = (snd(poss-maj(eq;cs;hd(cs)))) ∈ Cmd))
BY
{ ((InstLemma `new_23_sig_quorum_invariant_fun` [⌈Cmd⌉;⌈notify⌉;⌈propose⌉;⌈f⌉;⌈es.e1⌉;⌈e'⌉;⌈<z1, 0>⌉]⋅ THENA Auto)
THEN (InstLemma `new_23_sig_quorum_inv_vote_fun` [⌈Cmd⌉;⌈notify⌉;⌈propose⌉;⌈f⌉;⌈es.e1⌉;⌈e'⌉;⌈<z1, 0>⌉]⋅ THENA Auto)
THEN RepD
THEN D 0
THEN InstConcl [⌈loc(e')⌉;⌈[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;\000Ce')))]⌉;⌈[snd(msgval(e')) /
(snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1\000C;e')))]⌉]⋅
THEN RepeatFor 2 ((Reduce 0 THEN Auto))
THEN (RW ListC 0 THENA Auto)
THEN AutoSplit) }
1
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : ℕ
6. flrs : ℕ
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig{i:l}(ℤ)
10. f : new_23_sig_headers_type{i:l}(Cmd;notify;propose)
11. es : EO+(Message(f))
12. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. e : E@i
14. c : Cmd@i
15. n : ℤ@i
16. r : ℤ@i
17. <<n, r + 1>, c> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)@i
18. (f propose) = (ℤ × Cmd) ∈ Type
19. (f notify) = (ℤ × Cmd) ∈ Type
20. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
21. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
22. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
23. f ∈ Name ─→ Type
24. e' : E
25. delay : ℤ
26. (e' < e)
27. loc(e') ↓∈ reps
28. e1 : {e1:E| e1 ≤loc e' }
29. z1 : ℤ
30. z2 : Cmd
31. ((header(e1) = propose ∈ Name) ∧ has-es-info-type(es;e1;f;ℤ × Cmd) ∧ (<z1, z2> = msgval(e1) ∈ (ℤ × Cmd)))
∨ ((header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (z1 = (fst(fst(fst(msgval(e1))))) ∈ ℤ)
∧ (z2 = (snd(fst(msgval(e1)))) ∈ Cmd))
32. ¬↑(set-sig-member(slots) z1 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
33. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
34. (no new_23_sig_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, 0> between e1 and e')
35. header(e') = ``new_23_sig vote`` ∈ Name
36. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
37. <z1, 0> = (fst(fst(msgval(e')))) ∈ (ℤ × ℤ)
38. ¬(snd(msgval(e')) ∈ snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))
39. ||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = (coeff * flrs) ∈ ℤ
40. delay = 0 ∈ ℤ
41. ¬((fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))];\000Csnd(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ)
42. loc(e) = loc(e') ∈ Id
43. msg-authentic(info(e)) = ff
44. n = (fst(fst(fst(msgval(e'))))) ∈ ℤ
45. (r + 1) = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ
46. c = (snd(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))\000C];snd(fst(msgval(e')))))) ∈ Cmd
47. 0 ≤ (coeff * flrs)
48. no_repeats(Id;snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))
49. ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = ||fst(new_23_sig_QuorumStateFun(Cmd;noti\000Cfy;propose;f;<z1, 0>;es.e1;e'))|| ∈ ℤ
50. ∀i:ℕ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))||
(↓∃e'@0:E
((e'@0 <loc e')
∧ <<<z1, 0>, fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))[i]>, snd(new_23_sig_QuorumSt\000CateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))[i]> ∈
new_23_sig_vote'base(Cmd;notify;propose;f)(e'@0)))
51. no_repeats(Id;[snd(msgval(e')) / (snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))])
52. (||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) = ((coeff * flrs) + 1) ∈ ℤ
53. (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) = (||snd(new_23_sig_QuorumStateFun(C\000Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) ∈ ℤ
54. i : ℕ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1@i
55. msgval(e') ∈ ℤ × ℤ × Cmd × Id
56. i ≤ 0
⊢ ↓∃e'@0:E. mk-msg-interface(loc(e');make-Msg(``new_23_sig vote``;<<<n, r>, snd(fst(msgval(e')))>, snd(msgval(e'))>)) ∈ \000Cnew_23_sig_main()(e'@0)
2
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : ℕ
6. flrs : ℕ
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig{i:l}(ℤ)
10. f : new_23_sig_headers_type{i:l}(Cmd;notify;propose)
11. es : EO+(Message(f))
12. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. e : E@i
14. c : Cmd@i
15. n : ℤ@i
16. r : ℤ@i
17. <<n, r + 1>, c> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)@i
18. (f propose) = (ℤ × Cmd) ∈ Type
19. (f notify) = (ℤ × Cmd) ∈ Type
20. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
21. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
22. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
23. f ∈ Name ─→ Type
24. e' : E
25. delay : ℤ
26. (e' < e)
27. loc(e') ↓∈ reps
28. e1 : {e1:E| e1 ≤loc e' }
29. z1 : ℤ
30. z2 : Cmd
31. ((header(e1) = propose ∈ Name) ∧ has-es-info-type(es;e1;f;ℤ × Cmd) ∧ (<z1, z2> = msgval(e1) ∈ (ℤ × Cmd)))
∨ ((header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (z1 = (fst(fst(fst(msgval(e1))))) ∈ ℤ)
∧ (z2 = (snd(fst(msgval(e1)))) ∈ Cmd))
32. ¬↑(set-sig-member(slots) z1 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
33. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
34. (no new_23_sig_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, 0> between e1 and e')
35. header(e') = ``new_23_sig vote`` ∈ Name
36. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
37. <z1, 0> = (fst(fst(msgval(e')))) ∈ (ℤ × ℤ)
38. ¬(snd(msgval(e')) ∈ snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))
39. ||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = (coeff * flrs) ∈ ℤ
40. delay = 0 ∈ ℤ
41. ¬((fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))];\000Csnd(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ)
42. loc(e) = loc(e') ∈ Id
43. msg-authentic(info(e)) = ff
44. n = (fst(fst(fst(msgval(e'))))) ∈ ℤ
45. (r + 1) = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ
46. c = (snd(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))\000C];snd(fst(msgval(e')))))) ∈ Cmd
47. 0 ≤ (coeff * flrs)
48. no_repeats(Id;snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))
49. ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = ||fst(new_23_sig_QuorumStateFun(Cmd;noti\000Cfy;propose;f;<z1, 0>;es.e1;e'))|| ∈ ℤ
50. ∀i:ℕ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))||
(↓∃e'@0:E
((e'@0 <loc e')
∧ <<<z1, 0>, fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))[i]>, snd(new_23_sig_QuorumSt\000CateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))[i]> ∈
new_23_sig_vote'base(Cmd;notify;propose;f)(e'@0)))
51. no_repeats(Id;[snd(msgval(e')) / (snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))])
52. (||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) = ((coeff * flrs) + 1) ∈ ℤ
53. (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) = (||snd(new_23_sig_QuorumStateFun(C\000Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) ∈ ℤ
54. i : ℕ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1@i
55. ¬(i ≤ 0)
56. msgval(e') ∈ ℤ × ℤ × Cmd × Id
⊢ ↓∃e'@0:E
mk-msg-interface(loc(e');make-Msg(``new_23_sig vote``;<<<n, r>
, fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;\000Ces.e1;e'))[i - 1]
>
, snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;e\000Cs.e1;e'))[i - 1]
>)) ∈ new_23_sig_main()(e'@0)
3
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : ℕ
6. flrs : ℕ
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig{i:l}(ℤ)
10. f : new_23_sig_headers_type{i:l}(Cmd;notify;propose)
11. es : EO+(Message(f))
12. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. e : E@i
14. c : Cmd@i
15. n : ℤ@i
16. r : ℤ@i
17. <<n, r + 1>, c> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)@i
18. (f propose) = (ℤ × Cmd) ∈ Type
19. (f notify) = (ℤ × Cmd) ∈ Type
20. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
21. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
22. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
23. f ∈ Name ─→ Type
24. e' : E
25. delay : ℤ
26. (e' < e)
27. loc(e') ↓∈ reps
28. e1 : {e1:E| e1 ≤loc e' }
29. z1 : ℤ
30. z2 : Cmd
31. ((header(e1) = propose ∈ Name) ∧ has-es-info-type(es;e1;f;ℤ × Cmd) ∧ (<z1, z2> = msgval(e1) ∈ (ℤ × Cmd)))
∨ ((header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (z1 = (fst(fst(fst(msgval(e1))))) ∈ ℤ)
∧ (z2 = (snd(fst(msgval(e1)))) ∈ Cmd))
32. ¬↑(set-sig-member(slots) z1 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
33. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
34. (no new_23_sig_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, 0> between e1 and e')
35. header(e') = ``new_23_sig vote`` ∈ Name
36. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
37. <z1, 0> = (fst(fst(msgval(e')))) ∈ (ℤ × ℤ)
38. ¬(snd(msgval(e')) ∈ snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))
39. ||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = (coeff * flrs) ∈ ℤ
40. delay = 0 ∈ ℤ
41. ¬((fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))];\000Csnd(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ)
42. loc(e) = loc(e') ∈ Id
43. msg-authentic(info(e)) = ff
44. n = (fst(fst(fst(msgval(e'))))) ∈ ℤ
45. (r + 1) = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ
46. c = (snd(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))\000C];snd(fst(msgval(e')))))) ∈ Cmd
47. 0 ≤ (coeff * flrs)
48. no_repeats(Id;snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))
49. ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = ||fst(new_23_sig_QuorumStateFun(Cmd;noti\000Cfy;propose;f;<z1, 0>;es.e1;e'))|| ∈ ℤ
50. ∀i:ℕ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))||
(↓∃e'@0:E
((e'@0 <loc e')
∧ <<<z1, 0>, fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))[i]>, snd(new_23_sig_QuorumSt\000CateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))[i]> ∈
new_23_sig_vote'base(Cmd;notify;propose;f)(e'@0)))
51. no_repeats(Id;[snd(msgval(e')) / (snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))])
52. (||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) = ((coeff * flrs) + 1) ∈ ℤ
53. (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) = (||snd(new_23_sig_QuorumStateFun(C\000Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) ∈ ℤ
54. i : ℕ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1@i
55. ↓∃e'@0:E
mk-msg-interface(loc(e');make-Msg(``new_23_sig vote``;<<<n, r>
, [snd(fst(msgval(e'))) /
(fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1
, 0
>;es\000C.e1;e')))][i]
>
, [snd(msgval(e')) /
(snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1
, 0
>;es.\000Ce1;e')))][i]
>)) ∈ new_23_sig_main()(e'@0)
56. msgval(e') ∈ ℤ × ℤ × Cmd × Id
57. i ≤ 0
⊢ snd(msgval(e')) ↓∈ reps
4
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : ℕ
6. flrs : ℕ
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig{i:l}(ℤ)
10. f : new_23_sig_headers_type{i:l}(Cmd;notify;propose)
11. es : EO+(Message(f))
12. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. e : E@i
14. c : Cmd@i
15. n : ℤ@i
16. r : ℤ@i
17. <<n, r + 1>, c> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)@i
18. (f propose) = (ℤ × Cmd) ∈ Type
19. (f notify) = (ℤ × Cmd) ∈ Type
20. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
21. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
22. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
23. f ∈ Name ─→ Type
24. e' : E
25. delay : ℤ
26. (e' < e)
27. loc(e') ↓∈ reps
28. e1 : {e1:E| e1 ≤loc e' }
29. z1 : ℤ
30. z2 : Cmd
31. ((header(e1) = propose ∈ Name) ∧ has-es-info-type(es;e1;f;ℤ × Cmd) ∧ (<z1, z2> = msgval(e1) ∈ (ℤ × Cmd)))
∨ ((header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (z1 = (fst(fst(fst(msgval(e1))))) ∈ ℤ)
∧ (z2 = (snd(fst(msgval(e1)))) ∈ Cmd))
32. ¬↑(set-sig-member(slots) z1 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
33. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
34. (no new_23_sig_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, 0> between e1 and e')
35. header(e') = ``new_23_sig vote`` ∈ Name
36. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
37. <z1, 0> = (fst(fst(msgval(e')))) ∈ (ℤ × ℤ)
38. ¬(snd(msgval(e')) ∈ snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))
39. ||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = (coeff * flrs) ∈ ℤ
40. delay = 0 ∈ ℤ
41. ¬((fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))];\000Csnd(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ)
42. loc(e) = loc(e') ∈ Id
43. msg-authentic(info(e)) = ff
44. n = (fst(fst(fst(msgval(e'))))) ∈ ℤ
45. (r + 1) = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ
46. c = (snd(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))\000C];snd(fst(msgval(e')))))) ∈ Cmd
47. 0 ≤ (coeff * flrs)
48. no_repeats(Id;snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))
49. ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = ||fst(new_23_sig_QuorumStateFun(Cmd;noti\000Cfy;propose;f;<z1, 0>;es.e1;e'))|| ∈ ℤ
50. ∀i:ℕ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))||
(↓∃e'@0:E
((e'@0 <loc e')
∧ <<<z1, 0>, fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))[i]>, snd(new_23_sig_QuorumSt\000CateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))[i]> ∈
new_23_sig_vote'base(Cmd;notify;propose;f)(e'@0)))
51. no_repeats(Id;[snd(msgval(e')) / (snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))])
52. (||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) = ((coeff * flrs) + 1) ∈ ℤ
53. (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) = (||snd(new_23_sig_QuorumStateFun(C\000Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1) ∈ ℤ
54. i : ℕ||snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| + 1@i
55. ¬(i ≤ 0)
56. ↓∃e'@0:E
mk-msg-interface(loc(e');make-Msg(``new_23_sig vote``;<<<n, r>
, [snd(fst(msgval(e'))) /
(fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1
, 0
>;es\000C.e1;e')))][i]
>
, [snd(msgval(e')) /
(snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1
, 0
>;es.\000Ce1;e')))][i]
>)) ∈ new_23_sig_main()(e'@0)
57. msgval(e') ∈ ℤ × ℤ × Cmd × Id
⊢ snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))[i - 1] ↓∈ reps
Latex:
Latex:
1. Cmd : \{T:Type| valueall-type(T)\}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : \mBbbN{}
6. flrs : \mBbbN{}
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig\{i:l\}(\mBbbZ{})
10. f : new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose)
11. es : EO+(Message(f))
12. new\_23\_sig\_message-constraint\{i:l\}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. e : E@i
14. c : Cmd@i
15. n : \mBbbZ{}@i
16. r : \mBbbZ{}@i
17. <<n, r + 1>, c> \mmember{} new\_23\_sig\_retry'base(Cmd;notify;propose;f)(e)@i
18. (f propose) = (\mBbbZ{} \mtimes{} Cmd)
19. (f notify) = (\mBbbZ{} \mtimes{} Cmd)
20. (f ``new\_23\_sig decided``) = (\mBbbZ{} \mtimes{} Cmd)
21. (f ``new\_23\_sig retry``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd)
22. (f ``new\_23\_sig vote``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
23. f \mmember{} Name {}\mrightarrow{} Type
24. e' : E
25. delay : \mBbbZ{}
26. (e' < e)
27. loc(e') \mdownarrow{}\mmember{} reps
28. e1 : \{e1:E| e1 \mleq{}loc e' \}
29. z1 : \mBbbZ{}
30. z2 : Cmd
31. ((header(e1) = propose) \mwedge{} has-es-info-type(es;e1;f;\mBbbZ{} \mtimes{} Cmd) \mwedge{} (<z1, z2> = msgval(e1)))
\mvee{} ((header(e1) = ``new\_23\_sig vote``)
\mwedge{} has-es-info-type(es;e1;f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
\mwedge{} (z1 = (fst(fst(fst(msgval(e1))))))
\mwedge{} (z2 = (snd(fst(msgval(e1))))))
32. \mneg{}\muparrow{}(set-sig-member(slots) z1 new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
33. (no new\_23\_sig\_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
34. (no new\_23\_sig\_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, 0> between e1 and e')
35. header(e') = ``new\_23\_sig vote``
36. has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
37. <z1, 0> = (fst(fst(msgval(e'))))
38. \mneg{}(snd(msgval(e')) \mmember{} snd(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>es.e1;e')))
39. ||fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>es.e1;e'))|| = (coeff * flrs)
40. delay = 0
41. \mneg{}((fst(poss-maj(eq;[snd(fst(msgval(e'))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>es.e1;e')))];snd\000C(fst(msgval(e'))))))
= ((coeff * flrs) + 1))
42. loc(e) = loc(e')
43. msg-authentic(info(e)) = ff
44. n = (fst(fst(fst(msgval(e')))))
45. (r + 1) = ((snd(fst(fst(msgval(e'))))) + 1)
46. c
= (snd(poss-maj(eq;[snd(fst(msgval(e'))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>es.e1;e')))];snd(fst\000C(msgval(e'))))))
47. 0 \mleq{} (coeff * flrs)
\mvdash{} \mdownarrow{}\mexists{}x:Id
\mexists{}cs:Cmd List\msupplus{}
\mexists{}L:Id List
((no\_repeats(Id;L) \mwedge{} (||L|| = ((coeff * flrs) + 1)) \mwedge{} (||cs|| = ||L||))
\mwedge{} (\mforall{}i:\mBbbN{}||L||
((\mdownarrow{}\mexists{}e':E. mk-msg-interface(x;make-Msg(``new\_23\_sig vote``;<<<n, r>, cs[i]>, L[i]>)) \mmember{} new\000C\_23\_sig\_main()(e')) \mwedge{} L[i] \mdownarrow{}\mmember{} reps))
\mwedge{} (c = (snd(poss-maj(eq;cs;hd(cs))))))
By
Latex:
((InstLemma `new\_23\_sig\_quorum\_invariant\_fun` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}notify\mkleeneclose{};\mkleeneopen{}propose\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es.e1\mkleeneclose{};\mkleeneopen{}e'\mkleeneclose{};\mkleeneopen{}<z1, 0>\mkleeneclose{}]\mcdot{}
THENA Auto
)
THEN (InstLemma `new\_23\_sig\_quorum\_inv\_vote\_fun` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}notify\mkleeneclose{};\mkleeneopen{}propose\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es.e1\mkleeneclose{};\mkleeneopen{}e'\mkleeneclose{};\mkleeneopen{}<z1
, 0
>\mkleeneclose{}]\mcdot{}
THENA Auto
)
THEN RepD
THEN D 0
THEN InstConcl [\mkleeneopen{}loc(e')\mkleeneclose{};\mkleeneopen{}[snd(fst(msgval(e'))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>es.e1;e')))\000C]\mkleeneclose{};
\mkleeneopen{}[snd(msgval(e')) / (snd(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>es.e1;e')))]\mkleeneclose{}]\mcdot{}
THEN RepeatFor 2 ((Reduce 0 THEN Auto))
THEN (RW ListC 0 THENA Auto)
THEN AutoSplit)
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