Step
*
1
2
1
1
2
1
of Lemma
new_23_sig_progress-step10
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. e' : E
30. ((coeff * flrs) + 1)
= ||mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e'])||
∈ ℤ
31. e ≤loc e'
32. ↑new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0)
33. ¬↑(∃x∈[e, e'].new_23_sig_decided_with_id(Cmd;notify;propose;f;es;x;n))_b
34. 0 ≤ (coeff * flrs)
35. e ≤loc e'
36. ¬↑null(filter(λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);[e, e']))
37. ↑new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e';n;0)
38. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
39. poss-maj(eq;rev(mapfilter(λe.(snd(fst(msgval(e))));
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e']));snd(fst(msgval(e')))) ∈ ℤ × Cmd
40. e ≤loc e'
41. ¬((fst(poss-maj(eq;rev(mapfilter(λe.(snd(fst(msgval(e))));
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e']));snd(fst(msgval(e'))))))
= ((coeff * flrs) + 1)
∈ ℤ)
42. [e, e'] ∈ {x:E| e ≤loc x } List
⊢ mk-msg-interface(loc(e);make-Msg(``new_23_sig retry``;<<n, 1>
, snd(poss-maj(eq;rev(mapfilter(λe.(snd(fst(msgval(e))));
λe'....;
[e, e']));snd(fst(msgval(e')))))
>)) ∈ new_23_sig_main()(e')
BY
{ (RewriteWithILF 0 THEN Auto) }
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. e' : E
30. ((coeff * flrs) + 1)
= ||mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e'])||
∈ ℤ
31. e ≤loc e'
32. ↑new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0)
33. ¬↑(∃x∈[e, e'].new_23_sig_decided_with_id(Cmd;notify;propose;f;es;x;n))_b
34. 0 ≤ (coeff * flrs)
35. e ≤loc e'
36. ¬↑null(filter(λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);[e, e']))
37. ↑new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e';n;0)
38. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
39. poss-maj(eq;rev(mapfilter(λe.(snd(fst(msgval(e))));
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e']));snd(fst(msgval(e')))) ∈ ℤ × Cmd
40. e ≤loc e'
41. ¬((fst(poss-maj(eq;rev(mapfilter(λe.(snd(fst(msgval(e))));
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e']));snd(fst(msgval(e'))))))
= ((coeff * flrs) + 1)
∈ ℤ)
42. [e, e'] ∈ {x:E| e ≤loc x } List
43. loc(e') ↓∈ reps
⊢ ↓∃e1:{e1:E| e1 ≤loc e' }
∃z1:ℤ
∃z2:Cmd
(((((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)))
∧ (¬↑(set-sig-member(slots) z1 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))))
∧ (no new_23_sig_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
∧ (((no new_23_sig_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, 0> between e1 and e')
∧ ((header(e') = ``new_23_sig vote`` ∈ Name) ∧ has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id))
∧ (↑(new_23_sig_newvote(Cmd) <z1, 0> msgval(e') new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')\000C))
∧ (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = (coeff * flrs) ∈ ℤ)
∧ (0 = 0 ∈ ℤ)
∧ (¬((fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;\000Ce')))];snd(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ))
∧ (loc(e) = loc(e') ∈ Id)
∧ ff = ff
∧ ((n = (fst(fst(fst(msgval(e'))))) ∈ ℤ) ∧ (1 = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ))
∧ ((snd(poss-maj(eq;rev(mapfilter(λe.(snd(fst(msgval(e))));
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e']));snd(fst(msgval(e'))))))
= (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))
∨ (∃e2:{e1:E| e1 ≤loc e' }
∃z6:ℤ
∃z7:Cmd
(new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;z1;es.e1;e2) < z6
∧ (((header(e2) = ``new_23_sig retry`` ∈ Name)
∧ has-es-info-type(es;e2;f;ℤ × ℤ × Cmd)
∧ (<<z1, z6>, z7> = msgval(e2) ∈ (ℤ × ℤ × Cmd)))
∨ ((header(e2) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e2;f;ℤ × ℤ × Cmd × Id)
∧ (<<z1, z6>, z7> = (fst(msgval(e2))) ∈ (ℤ × ℤ × Cmd))))
∧ (no new_23_sig_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, z6> between e2 and e')
∧ ((header(e') = ``new_23_sig vote`` ∈ Name) ∧ has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id))
∧ (↑(new_23_sig_newvote(Cmd) <z1, z6> msgval(e') new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;e\000Cs.e2;e')))
∧ (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e2;e'))|| = (coeff * flrs) ∈ ℤ)
∧ (0 = 0 ∈ ℤ)
∧ (¬((fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>\000C;es.e2;e')))];snd(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ))
∧ (loc(e) = loc(e') ∈ Id)
∧ ff = ff
∧ ((n = (fst(fst(fst(msgval(e'))))) ∈ ℤ) ∧ (1 = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ))
∧ ((snd(poss-maj(eq;rev(mapfilter(λe.(snd(fst(msgval(e))));
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e']));snd(fst(msgval(e'))))))
= (snd(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;\000Ces.e2;e')))];snd(fst(msgval(e'))))))
∈ 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. e' : E
30. ((coeff * flrs) + 1)
= ||mapfilter(\mlambda{}e.<snd(msgval(e)), e, snd(fst(msgval(e)))>
\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e'])||
31. e \mleq{}loc e'
32. \muparrow{}new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0)
33. \mneg{}\muparrow{}(\mexists{}x\mmember{}[e, e'].new\_23\_sig\_decided\_with\_id(Cmd;notify;propose;f;es;x;n))\_b
34. 0 \mleq{} (coeff * flrs)
35. e \mleq{}loc e'
36. \mneg{}\muparrow{}null(filter(\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);[e, e']))
37. \muparrow{}new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e';n;0)
38. has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
39. poss-maj(eq;rev(mapfilter(\mlambda{}e.(snd(fst(msgval(e))));
...;
[e, e']));snd(fst(msgval(e')))) \mmember{} \mBbbZ{} \mtimes{} Cmd
40. e \mleq{}loc e'
41. \mneg{}((fst(poss-maj(eq;rev(mapfilter(\mlambda{}e.(snd(fst(msgval(e))));
...;
[e, e']));snd(fst(msgval(e'))))))
= ((coeff * flrs) + 1))
42. [e, e'] \mmember{} \{x:E| e \mleq{}loc x \} List
\mvdash{} mk-msg-interface(loc(e);make-Msg(``new\_23\_sig retry``;<<n, 1>
, snd(poss-maj(eq;rev(...);snd(fst(...))))
>)) \mmember{} new\_23\_sig\_main()(e')
By
Latex:
(RewriteWithILF 0 THEN Auto)
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