Step
*
2
2
of Lemma
pv11_p1_adopted_from_init_or_preempted
1. Cmd : Type@i'
2. valueall-type(Cmd)@i'
3. f : pv11_p1_headers_type{i:l}(Cmd)@i'
4. (f [decision]) = (ℤ × Cmd) ∈ Type
5. (f [propose]) = (ℤ × Cmd) ∈ Type
6. (f ``pv11_p1 adopted``) = (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)) ∈ Type
7. (f ``pv11_p1 preempted``) = pv11_p1_Ballot_Num() ∈ Type
8. (f ``pv11_p1 p2b``) = (Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num()) ∈ Type
9. (f ``pv11_p1 p2a``) = (Id × pv11_p1_Ballot_Num() × ℤ × Cmd) ∈ Type
10. (f ``pv11_p1 p1b``)
= (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
∈ Type
11. (f ``pv11_p1 p1a``) = (Id × pv11_p1_Ballot_Num()) ∈ Type
12. f ∈ Name ─→ Type
13. es : EO+(Message(f))@i'
14. e : E@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs_uid : Id ─→ ℤ@i
18. reps : bag(Id)@i
19. bnum : pv11_p1_Ballot_Num()@i
20. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
21. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
22. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)@i
23. e' : E
24. delay : ℤ
25. (e' < e)
26. loc(e') ↓∈ ldrs
27. e1 : E
28. e1 ≤loc e'
29. header(e1) = ``pv11_p1 preempted`` ∈ Name
30. has-es-info-type(es;e1;f;pv11_p1_Ballot_Num())
31. ↑(pv11_p1_is_bnum() msgval(e1))
32. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)) < msgval(e1))
33. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1)) o pv11_p1_p1b'base(Cmd;f)) o
pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1))) between e1 and e')
34. header(e') = ``pv11_p1 p1b`` ∈ Name
35. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
36. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
37. delay = 0 ∈ ℤ
38. loc(e) = loc(e') ∈ Id
39. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
40. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1)
loc(e1);es.e1;e'))) < pv11_p1_threshold(accpts)
41. msg-authentic(info(e)) = pv11_p1_init_active()
42. bnum = (pv11_p1_upd_bnum() msgval(e1) loc(e1)) ∈ pv11_p1_Ballot_Num()
43. pvals
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1) loc(e1);es.e1;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
44. (e1 <loc e)
45. msgval(e1) ∈ pv11_p1_preempted'base(Cmd;f)(e1)
⊢ <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))
, fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))
, snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)
BY
{ (BLemma `pv11_p1_LeaderState-classrel` THEN Auto THEN RepeatFor 2 ((RWO "pair-eta<" 0 THEN Auto))) }
Latex:
Latex:
1. Cmd : Type@i'
2. valueall-type(Cmd)@i'
3. f : pv11\_p1\_headers\_type\{i:l\}(Cmd)@i'
4. (f [decision]) = (\mBbbZ{} \mtimes{} Cmd)
5. (f [propose]) = (\mBbbZ{} \mtimes{} Cmd)
6. (f ``pv11\_p1 adopted``) = (pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
7. (f ``pv11\_p1 preempted``) = pv11\_p1\_Ballot\_Num()
8. (f ``pv11\_p1 p2b``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
9. (f ``pv11\_p1 p2a``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
10. (f ``pv11\_p1 p1b``)
= (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
11. (f ``pv11\_p1 p1a``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num())
12. f \mmember{} Name {}\mrightarrow{} Type
13. es : EO+(Message(f))@i'
14. e : E@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
18. reps : bag(Id)@i
19. bnum : pv11\_p1\_Ballot\_Num()@i
20. pvals : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
21. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
22. <bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e)@i
23. e' : E
24. delay : \mBbbZ{}
25. (e' < e)
26. loc(e') \mdownarrow{}\mmember{} ldrs
27. e1 : E
28. e1 \mleq{}loc e'
29. header(e1) = ``pv11\_p1 preempted``
30. has-es-info-type(es;e1;f;pv11\_p1\_Ballot\_Num())
31. \muparrow{}(pv11\_p1\_is\_bnum() msgval(e1))
32. \muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)) < msgval(e1))
33. (no ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) o
pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f)
(pv11\_p1\_upd\_bnum() msgval(e1) loc(e1))) between e1 and e')
34. header(e') = ``pv11\_p1 p1b``
35. has-es-info-type(es;e';f;Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
36. (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e'))))
37. delay = 0
38. loc(e) = loc(e')
39. (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e')))))
40. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1)
loc(e1);es.e1;e'))) < pv11\_p1\_threshold(accpts)
41. msg-authentic(info(e)) = pv11\_p1\_init\_active()
42. bnum = (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1))
43. pvals
= (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1) loc(e1);es.e1;e')))
44. (e1 <loc e)
45. msgval(e1) \mmember{} pv11\_p1\_preempted'base(Cmd;f)(e1)
\mvdash{} <fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1))
, fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)))
, snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)))> \mmember{} pv11\_p1\_Leader\000CState(Cmd;ldrs$_{uid}$;f)(e1)
By
Latex:
(BLemma `pv11\_p1\_LeaderState-classrel` THEN Auto THEN RepeatFor 2 ((RWO "pair-eta<" 0 THEN Auto)))
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