Step
*
1
of Lemma
new_23_sig_validity
.....assertion.....
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'
⊢ ∀e:E
((∀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')))))
∧ (∀v:ℤ × ℤ × Cmd × Id
(v ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e)
⇒ (↓∃e':E. ((e' < e) ∧ <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)(e)
⇒ (↓∃e':E. ((e' < e) ∧ <fst(fst(v)), snd(v)> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e')))))
∧ (∀e1:E
(e1 c≤ e
⇒ (∀n,rnd:ℤ. ∀c:Cmd.
((c ∈ fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<n, rnd>;es.e1;e)))
⇒ (↓∃e':E. ((e' < e) ∧ <n, c> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))))))))
BY
{ (CausalInd'
THEN Auto
THEN Try (DProdsAndUnions)
THEN Try (UseMessageConstraint (-1))
THEN SquashExRepD
THEN AllReduce) }
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. v1 : ℤ@i
22. v2 : Cmd@i
23. <v1, v2> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e)@i
24. e' : E
25. delay : ℤ
26. (e' < e)
27. loc(e') ↓∈ reps
28. e1 : {e1:E| e1 ≤loc e' }
29. z1 : ℤ
30. z2 : Cmd
31. ((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))
32. ¬↑(set-sig-member(slots) z1 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
33. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
34. ((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')))
∧ (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = (coeff * flrs) ∈ ℤ)
∧ (delay = 0 ∈ ℤ)
∧ ((fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))];snd\000C(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ)
∧ loc(e) ↓∈ reps
∧ msg-authentic(info(e)) = ff
∧ (v1 = (fst(fst(fst(msgval(e'))))) ∈ ℤ)
∧ (v2 = (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>;es.e2;e')\000C))
∧ (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e2;e'))|| = (coeff * flrs) ∈ ℤ)
∧ (delay = 0 ∈ ℤ)
∧ ((fst(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')))))) = ((coeff * flrs) + 1) ∈ ℤ)
∧ loc(e) ↓∈ reps
∧ msg-authentic(info(e)) = ff
∧ (v1 = (fst(fst(fst(msgval(e'))))) ∈ ℤ)
∧ (v2 = (snd(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e\000C2;e')))];snd(fst(msgval(e')))))) ∈ Cmd)))
⊢ ↓∃e':E. ((e' < e) ∧ <v1, v2> ∈ 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. ((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'))
3
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. ∀v:ℤ × ℤ × Cmd × Id
(v ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e)
⇒ (↓∃e':E. ((e' < e) ∧ <fst(fst(fst(v))), snd(fst(v))> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))))
23. v3 : ℤ@i
24. v4 : ℤ@i
25. v2 : Cmd@i
26. <<v3, v4>, v2> ∈ new_23_sig_retry'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. z1 : ℤ
33. z2 : Cmd
34. ((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))
35. ¬↑(set-sig-member(slots) z1 new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e1))
36. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) z1 between e1 and e')
37. ((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')))
∧ (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = (coeff * flrs) ∈ ℤ)
∧ (delay = 0 ∈ ℤ)
∧ (¬((fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))];s\000Cnd(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ))
∧ (loc(e) = loc(e') ∈ Id)
∧ msg-authentic(info(e)) = ff
∧ ((v3 = (fst(fst(fst(msgval(e'))))) ∈ ℤ) ∧ (v4 = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ))
∧ (v2 = (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>;es.e2;e')\000C))
∧ (||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e2;e'))|| = (coeff * flrs) ∈ ℤ)
∧ (delay = 0 ∈ ℤ)
∧ (¬((fst(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')))))) = ((coeff * flrs) + 1) ∈ ℤ))
∧ (loc(e) = loc(e') ∈ Id)
∧ msg-authentic(info(e)) = ff
∧ ((v3 = (fst(fst(fst(msgval(e'))))) ∈ ℤ) ∧ (v4 = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ))
∧ (v2 = (snd(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e\000C2;e')))];snd(fst(msgval(e')))))) ∈ Cmd)))
⊢ ↓∃e':E. ((e' < e) ∧ <v3, v2> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))
4
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. ∀v:ℤ × ℤ × Cmd × Id
(v ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e)
⇒ (↓∃e':E. ((e' < e) ∧ <fst(fst(fst(v))), snd(fst(v))> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))))
23. ∀v:ℤ × ℤ × Cmd
(v ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)
⇒ (↓∃e':E. ((e' < e) ∧ <fst(fst(v)), snd(v)> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))))
24. e1 : E@i
25. e1 c≤ e@i
26. n : ℤ@i
27. rnd : ℤ@i
28. c : Cmd@i
29. (c ∈ fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<n, rnd>;es.e1;e)))@i
⊢ ↓∃e':E. ((e' < e) ∧ <n, c> ∈ new_23_sig_propose'base(Cmd;notify;propose;f)(e'))
Latex:
Latex:
.....assertion.....
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'
\mvdash{} \mforall{}e:E
((\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')))))
\mwedge{} (\mforall{}v:\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id
(v \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(e)
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((e' < e)
\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)(e)
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((e' < e)
\mwedge{} <fst(fst(v)), snd(v)> \mmember{} new\_23\_sig\_propose'base(Cmd;notify;propose;f)(e')))))
\mwedge{} (\mforall{}e1:E
(e1 c\mleq{} e
{}\mRightarrow{} (\mforall{}n,rnd:\mBbbZ{}. \mforall{}c:Cmd.
((c \mmember{} fst(new\_23\_sig\_QuorumStateFun(Cmd;notify;propose;f;<n, rnd>es.e1;e)))
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((e' < e) \mwedge{} <n, c> \mmember{} new\_23\_sig\_propose'base(Cmd;notify;propose;f)(e'))))))))
By
Latex:
(CausalInd'
THEN Auto
THEN Try (DProdsAndUnions)
THEN Try (UseMessageConstraint (-1))
THEN SquashExRepD
THEN AllReduce)
Home
Index