Step
*
1
of Lemma
new_23_sig_progress2-step1
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. r : ℤ
21. c : Cmd
22. faulty : bag(Id)
23. e' : E
24. c' : Cmd
25. msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)@i
26. bag-no-repeats(Id;reps)@i
27. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
28. loc(e) ↓∈ reps@i
29. ¬loc(e) ↓∈ faulty@i
30. e' ≤loc e @i
31. ((header(e') = propose ∈ Name) ∧ <n, c'> = body(e'))
∨ (has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
∧ (header(e') = ``new_23_sig vote`` ∈ Name)
∧ (n = (fst(fst(fst(msgval(e'))))) ∈ ℤ)
∧ (c' = (snd(fst(msgval(e')))) ∈ Cmd))
32. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))@i
33. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) n between e' and e)@i
34. <<n, r>, c> ∈ new_23_sig_RoundInfo(Cmd;notify;propose;f)(e)@i
35. new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r@i
36. i : Id@i
37. i ↓∈ reps@i
38. loc(e) ↓∈ reps
⊢ ((r = 0 ∈ ℤ) ∧ (c = c' ∈ Cmd) ∧ (e = e' ∈ E))
∨ (new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r
∧ (((header(e) = ``new_23_sig retry`` ∈ Name)
∧ has-es-info-type(es;e;f;ℤ × ℤ × Cmd)
∧ (<<n, r>, c> = msgval(e) ∈ (ℤ × ℤ × Cmd)))
∨ ((header(e) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e;f;ℤ × ℤ × Cmd × Id)
∧ (<<n, r>, c> = (fst(msgval(e))) ∈ (ℤ × ℤ × Cmd)))))
BY
{ ((RWO "new_23_sig_round_info_classrel2" (-5) THENA Auto) THEN All Reduce) }
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. r : ℤ
21. c : Cmd
22. faulty : bag(Id)
23. e' : E
24. c' : Cmd
25. msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)@i
26. bag-no-repeats(Id;reps)@i
27. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
28. loc(e) ↓∈ reps@i
29. ¬loc(e) ↓∈ faulty@i
30. e' ≤loc e @i
31. ((header(e') = propose ∈ Name) ∧ <n, c'> = body(e'))
∨ (has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
∧ (header(e') = ``new_23_sig vote`` ∈ Name)
∧ (n = (fst(fst(fst(msgval(e'))))) ∈ ℤ)
∧ (c' = (snd(fst(msgval(e')))) ∈ Cmd))
32. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))@i
33. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) n between e' and e)@i
34. ((header(e) = ``new_23_sig retry`` ∈ Name) ∧ <<n, r>, c> = body(e))
∨ (has-es-info-type(es;e;f;ℤ × ℤ × Cmd × Id)
∧ (header(e) = ``new_23_sig vote`` ∈ Name)
∧ (n = (fst(fst(fst(msgval(e))))) ∈ ℤ)
∧ (r = (snd(fst(fst(msgval(e))))) ∈ ℤ)
∧ (c = (snd(fst(msgval(e)))) ∈ Cmd))
35. new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r@i
36. i : Id@i
37. i ↓∈ reps@i
38. loc(e) ↓∈ reps
⊢ ((r = 0 ∈ ℤ) ∧ (c = c' ∈ Cmd) ∧ (e = e' ∈ E))
∨ (new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r
∧ (((header(e) = ``new_23_sig retry`` ∈ Name)
∧ has-es-info-type(es;e;f;ℤ × ℤ × Cmd)
∧ (<<n, r>, c> = msgval(e) ∈ (ℤ × ℤ × Cmd)))
∨ ((header(e) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e;f;ℤ × ℤ × Cmd × Id)
∧ (<<n, r>, c> = (fst(msgval(e))) ∈ (ℤ × ℤ × 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. r : \mBbbZ{}
21. c : Cmd
22. faulty : bag(Id)
23. e' : E
24. c' : Cmd
25. msgs-interface-delivered-with-omissions(f;es;new\_23\_sig\_main();faulty;flrs;reps)@i
26. bag-no-repeats(Id;reps)@i
27. \#(reps) = ((coeff * flrs) + flrs + 1)@i
28. loc(e) \mdownarrow{}\mmember{} reps@i
29. \mneg{}loc(e) \mdownarrow{}\mmember{} faulty@i
30. e' \mleq{}loc e @i
31. ((header(e') = propose) \mwedge{} <n, c'> = body(e'))
\mvee{} (has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
\mwedge{} (header(e') = ``new\_23\_sig vote``)
\mwedge{} (n = (fst(fst(fst(msgval(e'))))))
\mwedge{} (c' = (snd(fst(msgval(e'))))))
32. \mneg{}\muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))@i
33. (no new\_23\_sig\_Notify(Cmd;clients;notify;propose;f) n between e' and e)@i
34. <<n, r>, c> \mmember{} new\_23\_sig\_RoundInfo(Cmd;notify;propose;f)(e)@i
35. new\_23\_sig\_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r@i
36. i : Id@i
37. i \mdownarrow{}\mmember{} reps@i
38. loc(e) \mdownarrow{}\mmember{} reps
\mvdash{} ((r = 0) \mwedge{} (c = c') \mwedge{} (e = e'))
\mvee{} (new\_23\_sig\_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r
\mwedge{} (((header(e) = ``new\_23\_sig retry``)
\mwedge{} has-es-info-type(es;e;f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd)
\mwedge{} (<<n, r>, c> = msgval(e)))
\mvee{} ((header(e) = ``new\_23\_sig vote``)
\mwedge{} has-es-info-type(es;e;f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
\mwedge{} (<<n, r>, c> = (fst(msgval(e)))))))
By
Latex:
((RWO "new\_23\_sig\_round\_info\_classrel2" (-5) THENA Auto) THEN All Reduce)
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