Step
*
1
2
of Lemma
new_23_sig_validity
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. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
19. e : E@i
20. ∀e1:E
((e1 < e)
⇒ ((∀v:ℤ × Cmd
(v ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e1)
⇒ (↓∃e':E. ((e' < e1) ∧ v ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))
∧ (∀v:ℤ × ℤ × Cmd × Id
(v ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e1)
⇒ (↓∃e':E
((e' < e1) ∧ <fst(fst(fst(v))), snd(fst(v))> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))
∧ (∀v:ℤ × ℤ × Cmd
(v ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e1)
⇒ (↓∃e':E. ((e' < e1) ∧ <fst(fst(v)), snd(v)> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))
∧ (∀e2:E
(e2 c≤ e1
⇒ (∀n,rnd:ℤ. ∀c:Cmd.
((c ∈ fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<n, rnd>;es.e2;e1)))
⇒ (↓∃e':E. ((e' < e1) ∧ <n, c> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))))))
21. ∀v:ℤ × Cmd
(v ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e)
⇒ (↓∃e':E. ((e' < e) ∧ v ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))))
22. v5 : ℤ@i
23. v6 : ℤ@i
24. v4 : Cmd@i
25. v2 : Id@i
26. <<<v5, v6>, v4>, v2> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e)@i
27. e' : E
28. delay : ℤ
29. (e' < e)
30. loc(e') ↓∈ reps
31. e1 : {e1:E| e1 ≤loc e' }
32. z2 : Cmd
33. ((header(e1) = propose ∈ Name) ∧ has-es-info-type(es;e1;f;ℤ × Cmd) ∧ (<v5, z2> = msgval(e1) ∈ (ℤ × Cmd)))
∨ ((header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (v5 = (fst(fst(fst(msgval(e1))))) ∈ ℤ)
∧ (z2 = (snd(fst(msgval(e1)))) ∈ Cmd))
34. ¬↑(set-sig-member(slots) v5 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
35. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) v5 between e1 and e')
36. loc(e) ↓∈ reps
37. delay = 0 ∈ ℤ
38. msg-authentic(info(e)) = ff
39. v2 = loc(e') ∈ Id
40. ((v6 = 0 ∈ ℤ) ∧ (v4 = z2 ∈ Cmd) ∧ (e' = e1 ∈ E))
∨ (new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;v5;es.e1;e') < v6
∧ (((header(e') = ``new_23_sig retry`` ∈ Name)
∧ has-es-info-type(es;e';f;ℤ × ℤ × Cmd)
∧ (<<v5, v6>, v4> = msgval(e') ∈ (ℤ × ℤ × Cmd)))
∨ ((header(e') = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
∧ (<<v5, v6>, v4> = (fst(msgval(e'))) ∈ (ℤ × ℤ × Cmd)))))
⊢ ↓∃e':E. ((e' < e) ∧ <v5, v4> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))
BY
{ (D (-1) THEN RepD) }
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. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
19. e : E@i
20. ∀e1:E
((e1 < e)
⇒ ((∀v:ℤ × Cmd
(v ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e1)
⇒ (↓∃e':E. ((e' < e1) ∧ v ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))
∧ (∀v:ℤ × ℤ × Cmd × Id
(v ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e1)
⇒ (↓∃e':E
((e' < e1) ∧ <fst(fst(fst(v))), snd(fst(v))> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))
∧ (∀v:ℤ × ℤ × Cmd
(v ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e1)
⇒ (↓∃e':E. ((e' < e1) ∧ <fst(fst(v)), snd(v)> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))
∧ (∀e2:E
(e2 c≤ e1
⇒ (∀n,rnd:ℤ. ∀c:Cmd.
((c ∈ fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<n, rnd>;es.e2;e1)))
⇒ (↓∃e':E. ((e' < e1) ∧ <n, c> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))))))
21. ∀v:ℤ × Cmd
(v ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e)
⇒ (↓∃e':E. ((e' < e) ∧ v ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))))
22. v5 : ℤ@i
23. v6 : ℤ@i
24. v4 : Cmd@i
25. v2 : Id@i
26. <<<v5, v6>, v4>, v2> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e)@i
27. e' : E
28. delay : ℤ
29. (e' < e)
30. loc(e') ↓∈ reps
31. e1 : {e1:E| e1 ≤loc e' }
32. z2 : Cmd
33. ((header(e1) = propose ∈ Name) ∧ has-es-info-type(es;e1;f;ℤ × Cmd) ∧ (<v5, z2> = msgval(e1) ∈ (ℤ × Cmd)))
∨ ((header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (v5 = (fst(fst(fst(msgval(e1))))) ∈ ℤ)
∧ (z2 = (snd(fst(msgval(e1)))) ∈ Cmd))
34. ¬↑(set-sig-member(slots) v5 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
35. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) v5 between e1 and e')
36. loc(e) ↓∈ reps
37. delay = 0 ∈ ℤ
38. msg-authentic(info(e)) = ff
39. v2 = loc(e') ∈ Id
40. v6 = 0 ∈ ℤ
41. v4 = z2 ∈ Cmd
42. e' = e1 ∈ E
⊢ ↓∃e':E. ((e' < e) ∧ <v5, v4> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))
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. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
19. e : E@i
20. ∀e1:E
((e1 < e)
⇒ ((∀v:ℤ × Cmd
(v ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e1)
⇒ (↓∃e':E. ((e' < e1) ∧ v ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))
∧ (∀v:ℤ × ℤ × Cmd × Id
(v ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e1)
⇒ (↓∃e':E
((e' < e1) ∧ <fst(fst(fst(v))), snd(fst(v))> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))
∧ (∀v:ℤ × ℤ × Cmd
(v ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e1)
⇒ (↓∃e':E. ((e' < e1) ∧ <fst(fst(v)), snd(v)> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))
∧ (∀e2:E
(e2 c≤ e1
⇒ (∀n,rnd:ℤ. ∀c:Cmd.
((c ∈ fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<n, rnd>;es.e2;e1)))
⇒ (↓∃e':E. ((e' < e1) ∧ <n, c> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))))))
21. ∀v:ℤ × Cmd
(v ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e)
⇒ (↓∃e':E. ((e' < e) ∧ v ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))))
22. v5 : ℤ@i
23. v6 : ℤ@i
24. v4 : Cmd@i
25. v2 : Id@i
26. <<<v5, v6>, v4>, v2> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e)@i
27. e' : E
28. delay : ℤ
29. (e' < e)
30. loc(e') ↓∈ reps
31. e1 : {e1:E| e1 ≤loc e' }
32. z2 : Cmd
33. ((header(e1) = propose ∈ Name) ∧ has-es-info-type(es;e1;f;ℤ × Cmd) ∧ (<v5, z2> = msgval(e1) ∈ (ℤ × Cmd)))
∨ ((header(e1) = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e1;f;ℤ × ℤ × Cmd × Id)
∧ (v5 = (fst(fst(fst(msgval(e1))))) ∈ ℤ)
∧ (z2 = (snd(fst(msgval(e1)))) ∈ Cmd))
34. ¬↑(set-sig-member(slots) v5 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
35. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) v5 between e1 and e')
36. loc(e) ↓∈ reps
37. delay = 0 ∈ ℤ
38. msg-authentic(info(e)) = ff
39. v2 = loc(e') ∈ Id
40. new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;v5;es.e1;e') < v6
41. ((header(e') = ``new_23_sig retry`` ∈ Name)
∧ has-es-info-type(es;e';f;ℤ × ℤ × Cmd)
∧ (<<v5, v6>, v4> = msgval(e') ∈ (ℤ × ℤ × Cmd)))
∨ ((header(e') = ``new_23_sig vote`` ∈ Name)
∧ has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
∧ (<<v5, v6>, v4> = (fst(msgval(e'))) ∈ (ℤ × ℤ × Cmd)))
⊢ ↓∃e':E. ((e' < e) ∧ <v5, v4> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))
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. new\_23\_sig\_message-constraint\{i:l\}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
19. e : E@i
20. \mforall{}e1:E
((e1 < e)
{}\mRightarrow{} ((\mforall{}v:\mBbbZ{} \mtimes{} Cmd
(v \mmember{} new\_23\_sig\_decided'base(Cmd;notify;propose;f)(e1)
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E. ((e' < e1) \mwedge{} v \mmember{} new\_23\_sig\_propose'base(Cmd;notify;propose;f)(e')))))
\mwedge{} (\mforall{}v:\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id
(v \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(e1)
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((e' < e1)
\mwedge{} <fst(fst(fst(v))), snd(fst(v))> \mmember{}
new\_23\_sig\_propose'base(Cmd;notify;propose;f)(e')))))
\mwedge{} (\mforall{}v:\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd
(v \mmember{} new\_23\_sig\_retry'base(Cmd;notify;propose;f)(e1)
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((e' < e1)
\mwedge{} <fst(fst(v)), snd(v)> \mmember{} new\_23\_sig\_propose'base(Cmd;notify;propose;f)(e')))))
\mwedge{} (\mforall{}e2:E
(e2 c\mleq{} e1
{}\mRightarrow{} (\mforall{}n,rnd:\mBbbZ{}. \mforall{}c:Cmd.
((c \mmember{} fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<n, rnd>es.e2;e1)))
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((e' < e1)
\mwedge{} <n, c> \mmember{} new\_23\_sig\_propose'base(Cmd;notify;propose;f)(e')))))))))
21. \mforall{}v:\mBbbZ{} \mtimes{} Cmd
(v \mmember{} new\_23\_sig\_decided'base(Cmd;notify;propose;f)(e)
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E. ((e' < e) \mwedge{} v \mmember{} new\_23\_sig\_propose'base(Cmd;notify;propose;f)(e'))))
22. v5 : \mBbbZ{}@i
23. v6 : \mBbbZ{}@i
24. v4 : Cmd@i
25. v2 : Id@i
26. <<<v5, v6>, v4>, v2> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(e)@i
27. e' : E
28. delay : \mBbbZ{}
29. (e' < e)
30. loc(e') \mdownarrow{}\mmember{} reps
31. e1 : \{e1:E| e1 \mleq{}loc e' \}
32. z2 : Cmd
33. ((header(e1) = propose) \mwedge{} has-es-info-type(es;e1;f;\mBbbZ{} \mtimes{} Cmd) \mwedge{} (<v5, 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{} (v5 = (fst(fst(fst(msgval(e1))))))
\mwedge{} (z2 = (snd(fst(msgval(e1))))))
34. \mneg{}\muparrow{}(set-sig-member(slots) v5 new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
35. (no new\_23\_sig\_Notify(Cmd;clients;notify;propose;f) v5 between e1 and e')
36. loc(e) \mdownarrow{}\mmember{} reps
37. delay = 0
38. msg-authentic(info(e)) = ff
39. v2 = loc(e')
40. ((v6 = 0) \mwedge{} (v4 = z2) \mwedge{} (e' = e1))
\mvee{} (new\_23\_sig\_NewRoundsStateFun(Cmd;notify;propose;f;v5;es.e1;e') < v6
\mwedge{} (((header(e') = ``new\_23\_sig retry``)
\mwedge{} has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd)
\mwedge{} (<<v5, v6>, v4> = 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{} (<<v5, v6>, v4> = (fst(msgval(e')))))))
\mvdash{} \mdownarrow{}\mexists{}e':E. ((e' < e) \mwedge{} <v5, v4> \mmember{} new\_23\_sig\_propose'base(Cmd;notify;propose;f)(e'))
By
Latex:
(D (-1) THEN RepD)
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