Step
*
2
1
1
of Lemma
new_23_sig_decided_property
.....set predicate.....
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : ℤ
6. flrs : ℤ
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. <n, c> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e)@i
17. (f propose) = (ℤ × Cmd) ∈ Type
18. (f notify) = (ℤ × Cmd) ∈ Type
19. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
20. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
21. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
22. f ∈ Name ─→ Type
23. e' : E
24. delay : ℤ
25. (e' < e)
26. loc(e') ↓∈ reps
27. e1 : {e1:E| e1 ≤loc e' }
28. z1 : ℤ
29. z2 : Cmd
30. ((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))
31. ¬↑(set-sig-member(slots) z1 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
32. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
33. e2 : {e1:E| e1 ≤loc e' }
34. z6 : ℤ
35. z7 : Cmd
36. new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;z1;es.e1;e2) < z6
37. ((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)))
38. (no new_23_sig_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, z6> between e2 and e')
39. header(e') = ``new_23_sig vote`` ∈ Name
40. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
41. ↑(new_23_sig_newvote(Cmd) <z1, z6> msgval(e') new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e2;e'))
42. ||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e2;e'))|| = (coeff * flrs) ∈ ℤ
43. delay = 0 ∈ ℤ
44. (fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e2;e')))];s\000Cnd(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ
45. loc(e) ↓∈ reps
46. msg-authentic(info(e)) = ff
47. n = (fst(fst(fst(msgval(e'))))) ∈ ℤ
48. c = (snd(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e2;e'))\000C)];snd(fst(msgval(e')))))) ∈ Cmd
⊢ 0 ≤ ((coeff * flrs) + 1)
BY
{ (SubstFor ⌈coeff * flrs⌉ 0⋅ THEN All Thin THEN Auto') }
Latex:
Latex:
.....set predicate.....
1. Cmd : \{T:Type| valueall-type(T)\}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : \mBbbZ{}
6. flrs : \mBbbZ{}
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. <n, c> \mmember{} new\_23\_sig\_decided'base(Cmd;notify;propose;f)(e)@i
17. (f propose) = (\mBbbZ{} \mtimes{} Cmd)
18. (f notify) = (\mBbbZ{} \mtimes{} Cmd)
19. (f ``new\_23\_sig decided``) = (\mBbbZ{} \mtimes{} Cmd)
20. (f ``new\_23\_sig retry``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd)
21. (f ``new\_23\_sig vote``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
22. f \mmember{} Name {}\mrightarrow{} Type
23. e' : E
24. delay : \mBbbZ{}
25. (e' < e)
26. loc(e') \mdownarrow{}\mmember{} reps
27. e1 : \{e1:E| e1 \mleq{}loc e' \}
28. z1 : \mBbbZ{}
29. z2 : Cmd
30. ((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))))))
31. \mneg{}\muparrow{}(set-sig-member(slots) z1 new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
32. (no new\_23\_sig\_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
33. e2 : \{e1:E| e1 \mleq{}loc e' \}
34. z6 : \mBbbZ{}
35. z7 : Cmd
36. new\_23\_sig\_NewRoundsStateFun(Cmd;notify;propose;f;z1;es.e1;e2) < z6
37. ((header(e2) = ``new\_23\_sig retry``)
\mwedge{} has-es-info-type(es;e2;f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd)
\mwedge{} (<<z1, z6>, z7> = msgval(e2)))
\mvee{} ((header(e2) = ``new\_23\_sig vote``)
\mwedge{} has-es-info-type(es;e2;f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
\mwedge{} (<<z1, z6>, z7> = (fst(msgval(e2)))))
38. (no new\_23\_sig\_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, z6> between e2 and e')
39. header(e') = ``new\_23\_sig vote``
40. has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
41. \muparrow{}(new\_23\_sig\_newvote(Cmd) <z1, z6> msgval(e') new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1\000C, z6>es.e2;e'))
42. ||fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>es.e2;e'))|| = (coeff * flrs)
43. delay = 0
44. (fst(poss-maj(eq;[snd(fst(msgval(e'))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>es.e2;e')))];snd(\000Cfst(msgval(e'))))))
= ((coeff * flrs) + 1)
45. loc(e) \mdownarrow{}\mmember{} reps
46. msg-authentic(info(e)) = ff
47. n = (fst(fst(fst(msgval(e')))))
48. c
= (snd(poss-maj(eq;[snd(fst(msgval(e'))) /
(fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>es.e2;e')))];snd(fs\000Ct(msgval(e'))))))
\mvdash{} 0 \mleq{} ((coeff * flrs) + 1)
By
Latex:
(SubstFor \mkleeneopen{}coeff * flrs\mkleeneclose{} 0\mcdot{} THEN All Thin THEN Auto')
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