Step
*
of Lemma
new_23_sig_retry_property
∀[Cmd:ValueAllType]. ∀[eq:EqDecider(Cmd)]. ∀[reps,clients:bag(Id)]. ∀[coeff,flrs:ℕ]. ∀[propose,notify:Atom List].
∀[slots:set-sig{i:l}(ℤ)]. ∀[f:new_23_sig_headers_type{i:l}(Cmd;notify;propose)]. ∀[es:EO+(Message(f))].
(new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)
⇒ (∀e:E. ∀c:Cmd. ∀n,r:ℤ.
(<<n, r + 1>, c> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)
⇒ (↓∃x:Id
∃cs:Cmd List+
∃L:Id List
((no_repeats(Id;L) ∧ (||L|| = ((coeff * flrs) + 1) ∈ ℤ) ∧ (||cs|| = ||L|| ∈ ℤ))
∧ (∀i:ℕ||L||. ((↓∃e':E. mk-msg-interface(x;make-Msg(``new_23_sig vote``;<<<n, r>, cs[i]>, L[i]>)) ∈ new_\000C23_sig_main()(e')) ∧ L[i] ↓∈ reps))
∧ (c = (snd(poss-maj(eq;cs;hd(cs)))) ∈ Cmd))))))
BY
{ (Unfold `vatype` 0
THEN Auto
THEN AddMessageHeaderTypes
THEN UseMessageConstraint (-3)
THEN SquashExRepD
THEN D (-1)
THEN ExRepD
THEN All(\i.(RWO "new_23_sig_assert_newvote" i THENA Auto))⋅
THEN RepD
THEN (InstLemma `mul_bounds_1a` [⌈coeff⌉;⌈flrs⌉]⋅ THENA Auto)) }
1
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : ℕ
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. 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. r : ℤ@i
17. <<n, r + 1>, c> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)@i
18. (f propose) = (ℤ × Cmd) ∈ Type
19. (f notify) = (ℤ × Cmd) ∈ Type
20. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
21. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
22. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
23. f ∈ Name ─→ Type
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')
35. header(e') = ``new_23_sig vote`` ∈ Name
36. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
37. <z1, 0> = (fst(fst(msgval(e')))) ∈ (ℤ × ℤ)
38. ¬(snd(msgval(e')) ∈ snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))
39. ||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e'))|| = (coeff * flrs) ∈ ℤ
40. delay = 0 ∈ ℤ
41. ¬((fst(poss-maj(eq;[snd(fst(msgval(e'))) / (fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, 0>;es.e1;e')))];\000Csnd(fst(msgval(e')))))) = ((coeff * flrs) + 1) ∈ ℤ)
42. loc(e) = loc(e') ∈ Id
43. msg-authentic(info(e)) = ff
44. n = (fst(fst(fst(msgval(e'))))) ∈ ℤ
45. (r + 1) = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ
46. c = (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
47. 0 ≤ (coeff * flrs)
⊢ ↓∃x:Id
∃cs:Cmd List+
∃L:Id List
((no_repeats(Id;L) ∧ (||L|| = ((coeff * flrs) + 1) ∈ ℤ) ∧ (||cs|| = ||L|| ∈ ℤ))
∧ (∀i:ℕ||L||. ((↓∃e':E. mk-msg-interface(x;make-Msg(``new_23_sig vote``;<<<n, r>, cs[i]>, L[i]>)) ∈ new_23_sig_mai\000Cn()(e')) ∧ L[i] ↓∈ reps))
∧ (c = (snd(poss-maj(eq;cs;hd(cs)))) ∈ Cmd))
2
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : ℕ
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. 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. r : ℤ@i
17. <<n, r + 1>, c> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e)@i
18. (f propose) = (ℤ × Cmd) ∈ Type
19. (f notify) = (ℤ × Cmd) ∈ Type
20. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
21. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
22. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
23. f ∈ Name ─→ Type
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. e2 : {e1:E| e1 ≤loc e' }
35. z6 : ℤ
36. z7 : Cmd
37. new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;z1;es.e1;e2) < z6
38. ((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)))
39. (no new_23_sig_Quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <z1, z6> between e2 and e')
40. header(e') = ``new_23_sig vote`` ∈ Name
41. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
42. <z1, z6> = (fst(fst(msgval(e')))) ∈ (ℤ × ℤ)
43. ¬(snd(msgval(e')) ∈ snd(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e2;e')))
44. ||fst(new_23_sig_QuorumStateFun(Cmd;notify;propose;f;<z1, z6>;es.e2;e'))|| = (coeff * flrs) ∈ ℤ
45. delay = 0 ∈ ℤ
46. ¬((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)
47. loc(e) = loc(e') ∈ Id
48. msg-authentic(info(e)) = ff
49. n = (fst(fst(fst(msgval(e'))))) ∈ ℤ
50. (r + 1) = ((snd(fst(fst(msgval(e'))))) + 1) ∈ ℤ
51. 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
52. 0 ≤ (coeff * flrs)
⊢ ↓∃x:Id
∃cs:Cmd List+
∃L:Id List
((no_repeats(Id;L) ∧ (||L|| = ((coeff * flrs) + 1) ∈ ℤ) ∧ (||cs|| = ||L|| ∈ ℤ))
∧ (∀i:ℕ||L||. ((↓∃e':E. mk-msg-interface(x;make-Msg(``new_23_sig vote``;<<<n, r>, cs[i]>, L[i]>)) ∈ new_23_sig_mai\000Cn()(e')) ∧ L[i] ↓∈ reps))
∧ (c = (snd(poss-maj(eq;cs;hd(cs)))) ∈ Cmd))
Latex:
Latex:
\mforall{}[Cmd:ValueAllType]. \mforall{}[eq:EqDecider(Cmd)]. \mforall{}[reps,clients:bag(Id)]. \mforall{}[coeff,flrs:\mBbbN{}].
\mforall{}[propose,notify:Atom List]. \mforall{}[slots:set-sig\{i:l\}(\mBbbZ{})].
\mforall{}[f:new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose)]. \mforall{}[es:EO+(Message(f))].
(new\_23\_sig\_message-constraint\{i:l\}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)
{}\mRightarrow{} (\mforall{}e:E. \mforall{}c:Cmd. \mforall{}n,r:\mBbbZ{}.
(<<n, r + 1>, c> \mmember{} new\_23\_sig\_retry'base(Cmd;notify;propose;f)(e)
{}\mRightarrow{} (\mdownarrow{}\mexists{}x:Id
\mexists{}cs:Cmd List\msupplus{}
\mexists{}L:Id List
((no\_repeats(Id;L) \mwedge{} (||L|| = ((coeff * flrs) + 1)) \mwedge{} (||cs|| = ||L||))
\mwedge{} (\mforall{}i:\mBbbN{}||L||
((\mdownarrow{}\mexists{}e':E
mk-msg-interface(x;make-Msg(``new\_23\_sig vote``;<<<n, r>, cs[i]>, L[i]>)) \mmember{}\000C new\_23\_sig\_main()(e'))
\mwedge{} L[i] \mdownarrow{}\mmember{} reps))
\mwedge{} (c = (snd(poss-maj(eq;cs;hd(cs))))))))))
By
Latex:
(Unfold `vatype` 0
THEN Auto
THEN AddMessageHeaderTypes
THEN UseMessageConstraint (-3)
THEN SquashExRepD
THEN D (-1)
THEN ExRepD
THEN All(\mbackslash{}i.(RWO "new\_23\_sig\_assert\_newvote" i THENA Auto))\mcdot{}
THEN RepD
THEN (InstLemma `mul\_bounds\_1a` [\mkleeneopen{}coeff\mkleeneclose{};\mkleeneopen{}flrs\mkleeneclose{}]\mcdot{} THENA Auto))
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