Step
*
1
1
1
1
of Lemma
new_23_sig_progress-step3
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : {2...}
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. (f propose) = (ℤ × Cmd) ∈ Type
12. (f notify) = (ℤ × Cmd) ∈ Type
13. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
14. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
15. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
16. f ∈ Name ─→ Type
17. es : EO+(Message(f))
18. e : E
19. n : ℤ
20. c : Cmd
21. faulty : bag(Id)
22. msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)@i
23. bag-no-repeats(Id;reps)@i
24. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
25. loc(e) ↓∈ reps@i
26. ¬loc(e) ↓∈ faulty@i
27. <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)@i
28. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
29. ∀i:Id
(i ↓∈ reps
⇒ (↓∃e':E
((e < e') ∧ (loc(e') = i ∈ Id) ∧ <<<n, 0>, c>, loc(e)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e'))))
30. i : Id@i
31. i ↓∈ reps@i
32. e' : E
33. (e < e')
34. loc(e') = i ∈ Id
35. <<<n, 0>, c>, loc(e)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e')
36. ↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))
37. e1 : E
38. c1 : Cmd
39. (e1 <loc e')
40. <n, c1> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e1)
∨ <n, c1> ∈ (new_23_sig_vote2prop(Cmd) o new_23_sig_vote'base(Cmd;notify;propose;f))(e1)
41. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
42. loc(e1) = i ∈ Id
43. loc(e1) ↓∈ reps
⊢ ((header(e1) = propose ∈ Name) ∧ has-es-info-type(es;e1;f;ℤ × Cmd) ∧ (<n, c1> = msgval(e1) ∈ (ℤ × Cmd)))
∨ ((header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (n = (fst(fst(fst(msgval(e1))))) ∈ ℤ)
∧ (c1 = (snd(fst(msgval(e1)))) ∈ Cmd))
BY
{ (RepUR ``equal-info-body`` 0 THEN ParallelOp -4 THEN UseClassRel (-4)) }
1
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : {2...}
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. (f propose) = (ℤ × Cmd) ∈ Type
12. (f notify) = (ℤ × Cmd) ∈ Type
13. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
14. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
15. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
16. f ∈ Name ─→ Type
17. es : EO+(Message(f))
18. e : E
19. n : ℤ
20. c : Cmd
21. faulty : bag(Id)
22. msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)@i
23. bag-no-repeats(Id;reps)@i
24. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
25. loc(e) ↓∈ reps@i
26. ¬loc(e) ↓∈ faulty@i
27. <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)@i
28. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
29. ∀i:Id
(i ↓∈ reps
⇒ (↓∃e':E
((e < e') ∧ (loc(e') = i ∈ Id) ∧ <<<n, 0>, c>, loc(e)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e'))))
30. i : Id@i
31. i ↓∈ reps@i
32. e' : E
33. (e < e')
34. loc(e') = i ∈ Id
35. <<<n, 0>, c>, loc(e)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e')
36. ↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))
37. e1 : E
38. c1 : Cmd
39. (e1 <loc e')
40. header(e1) = propose ∈ Name
41. has-es-info-type(es;e1;f;ℤ × Cmd)
42. <n, c1> = msgval(e1) ∈ (ℤ × Cmd)
43. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
44. loc(e1) = i ∈ Id
45. loc(e1) ↓∈ reps
⊢ (header(e1) = propose ∈ Name) ∧ has-es-info-type(es;e1;f;ℤ × Cmd) ∧ (<n, c1> = msgval(e1) ∈ (ℤ × Cmd))
2
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : {2...}
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. (f propose) = (ℤ × Cmd) ∈ Type
12. (f notify) = (ℤ × Cmd) ∈ Type
13. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
14. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
15. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
16. f ∈ Name ─→ Type
17. es : EO+(Message(f))
18. e : E
19. n : ℤ
20. c : Cmd
21. faulty : bag(Id)
22. msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)@i
23. bag-no-repeats(Id;reps)@i
24. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
25. loc(e) ↓∈ reps@i
26. ¬loc(e) ↓∈ faulty@i
27. <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)@i
28. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
29. ∀i:Id
(i ↓∈ reps
⇒ (↓∃e':E
((e < e') ∧ (loc(e') = i ∈ Id) ∧ <<<n, 0>, c>, loc(e)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e'))))
30. i : Id@i
31. i ↓∈ reps@i
32. e' : E
33. (e < e')
34. loc(e') = i ∈ Id
35. <<<n, 0>, c>, loc(e)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e')
36. ↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))
37. e1 : E
38. c1 : Cmd
39. (e1 <loc e')
40. ↓∃b:ℤ × ℤ × Cmd × Id
(b ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e1) ∧ <n, c1> ↓∈ new_23_sig_vote2prop(Cmd) loc(e1) b)
41. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
42. loc(e1) = i ∈ Id
43. loc(e1) ↓∈ reps
⊢ (header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (n = (fst(fst(fst(msgval(e1))))) ∈ ℤ)
∧ (c1 = (snd(fst(msgval(e1)))) ∈ Cmd)
Latex:
Latex:
1. Cmd : \{T:Type| valueall-type(T)\}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : \{2...\}
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. (f propose) = (\mBbbZ{} \mtimes{} Cmd)
12. (f notify) = (\mBbbZ{} \mtimes{} Cmd)
13. (f ``new\_23\_sig decided``) = (\mBbbZ{} \mtimes{} Cmd)
14. (f ``new\_23\_sig retry``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd)
15. (f ``new\_23\_sig vote``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
16. f \mmember{} Name {}\mrightarrow{} Type
17. es : EO+(Message(f))
18. e : E
19. n : \mBbbZ{}
20. c : Cmd
21. faulty : bag(Id)
22. msgs-interface-delivered-with-omissions(f;es;new\_23\_sig\_main();faulty;flrs;reps)@i
23. bag-no-repeats(Id;reps)@i
24. \#(reps) = ((coeff * flrs) + flrs + 1)@i
25. loc(e) \mdownarrow{}\mmember{} reps@i
26. \mneg{}loc(e) \mdownarrow{}\mmember{} faulty@i
27. <n, c> \mmember{} new\_23\_sig\_Proposal(Cmd;notify;propose;f)(e)@i
28. \mneg{}\muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
29. \mforall{}i:Id
(i \mdownarrow{}\mmember{} reps
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((e < e')
\mwedge{} (loc(e') = i)
\mwedge{} <<<n, 0>, c>, loc(e)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(e'))))
30. i : Id@i
31. i \mdownarrow{}\mmember{} reps@i
32. e' : E
33. (e < e')
34. loc(e') = i
35. <<<n, 0>, c>, loc(e)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(e')
36. \muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))
37. e1 : E
38. c1 : Cmd
39. (e1 <loc e')
40. <n, c1> \mmember{} new\_23\_sig\_propose'base(Cmd;notify;propose;f)(e1)
\mvee{} <n, c1> \mmember{} (new\_23\_sig\_vote2prop(Cmd) o new\_23\_sig\_vote'base(Cmd;notify;propose;f))(e1)
41. \mneg{}\muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
42. loc(e1) = i
43. loc(e1) \mdownarrow{}\mmember{} reps
\mvdash{} ((header(e1) = propose) \mwedge{} has-es-info-type(es;e1;f;\mBbbZ{} \mtimes{} Cmd) \mwedge{} (<n, c1> = 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{} (n = (fst(fst(fst(msgval(e1))))))
\mwedge{} (c1 = (snd(fst(msgval(e1))))))
By
Latex:
(RepUR ``equal-info-body`` 0 THEN ParallelOp -4 THEN UseClassRel (-4))
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