Step
*
2
1
1
of Lemma
pv11_p1_adopted_from_acceptor
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. b : pv11_p1_Ballot_Num()@i
22. s : ℤ@i
23. c : Cmd@i
24. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
25. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)@i
26. (<b, s, c> ∈ pvals)@i
27. e' : E
28. delay : ℤ
29. (e' < e)
30. loc(e') ↓∈ ldrs
31. e1 : E
32. e1 ≤loc e'
33. header(e1) = ``pv11_p1 preempted`` ∈ Name
34. has-es-info-type(es;e1;f;pv11_p1_Ballot_Num())
35. ↑(pv11_p1_is_bnum() msgval(e1))
36. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)) < msgval(e1))
37. (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')
38. header(e') = ``pv11_p1 p1b`` ∈ Name
39. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
40. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
41. delay = 0 ∈ ℤ
42. loc(e) = loc(e') ∈ Id
43. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
44. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1)
loc(e1);es.e1;e'))) < pv11_p1_threshold(accpts)
45. msg-authentic(info(e)) = pv11_p1_init_active()
46. bnum = (pv11_p1_upd_bnum() msgval(e1) loc(e1)) ∈ pv11_p1_Ballot_Num()
47. 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)
48. ∀p:pv11_p1_Ballot_Num() × ℤ × Cmd
((p ∈ pvals)
⇒ (∃e'@0:E
∃l:Id
∃r:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
(e'@0 ≤loc e'
∧ <l, pv11_p1_upd_bnum() msgval(e1) loc(e1), pv11_p1_upd_bnum() msgval(e1) loc(e1), r> ∈
pv11_p1_p1b'base(Cmd;f)(e'@0)
∧ (p ∈ r))))
49. e2 : E
50. l : Id
51. r : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
52. e2 ≤loc e'
53. <l, pv11_p1_upd_bnum() msgval(e1) loc(e1), pv11_p1_upd_bnum() msgval(e1) loc(e1), r> ∈ Base(``pv11_p1 p1b``)(e2)
54. (<b, s, c> ∈ r)
55. e3 : E
56. d1 : ℤ
57. (e3 < e2)
58. loc(e3) ↓∈ accpts
59. header(e3) = ``pv11_p1 p1a`` ∈ Name
60. has-es-info-type(es;e3;f;Id × pv11_p1_Ballot_Num())
61. d1 = 0 ∈ ℤ
62. loc(e2) = (fst(msgval(e3))) ∈ Id
63. msg-authentic(info(e2)) = pv11_p1_init_active()
64. l = loc(e3) ∈ Id
65. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (snd(msgval(e3))) ∈ pv11_p1_Ballot_Num()
66. <pv11_p1_upd_bnum() msgval(e1) loc(e1), r>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e3)
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⊢ ↓∃e':E
∃accepted:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
((e' < e) ∧ <bnum, accepted> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e') ∧ (<b, s, c> ∈ accepted))
BY
{ (D 0 THEN InstConcl [⌜e3⌝;⌜r⌝]⋅ THEN Auto THEN BLemma `pv11_p1_AcceptorState-classrel` 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. b : pv11\_p1\_Ballot\_Num()@i
22. s : \mBbbZ{}@i
23. c : Cmd@i
24. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
25. <bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e)@i
26. (<b, s, c> \mmember{} pvals)@i
27. e' : E
28. delay : \mBbbZ{}
29. (e' < e)
30. loc(e') \mdownarrow{}\mmember{} ldrs
31. e1 : E
32. e1 \mleq{}loc e'
33. header(e1) = ``pv11\_p1 preempted``
34. has-es-info-type(es;e1;f;pv11\_p1\_Ballot\_Num())
35. \muparrow{}(pv11\_p1\_is\_bnum() msgval(e1))
36. \muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)) < msgval(e1))
37. (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')
38. header(e') = ``pv11\_p1 p1b``
39. 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))
40. (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e'))))
41. delay = 0
42. loc(e) = loc(e')
43. (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e')))))
44. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1)
loc(e1);es.e1;e'))) < pv11\_p1\_threshold(accpts)
45. msg-authentic(info(e)) = pv11\_p1\_init\_active()
46. bnum = (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1))
47. pvals
= (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1) loc(e1);es.e1;e')))
48. \mforall{}p:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd
((p \mmember{} pvals)
{}\mRightarrow{} (\mexists{}e'@0:E
\mexists{}l:Id
\mexists{}r:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
(e'@0 \mleq{}loc e'
\mwedge{} <l
, pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)
, pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)
, r> \mmember{} pv11\_p1\_p1b'base(Cmd;f)(e'@0)
\mwedge{} (p \mmember{} r))))
49. e2 : E
50. l : Id
51. r : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
52. e2 \mleq{}loc e'
53. <l, pv11\_p1\_upd\_bnum() msgval(e1) loc(e1), pv11\_p1\_upd\_bnum() msgval(e1) loc(e1), r> \mmember{}
Base(``pv11\_p1 p1b``)(e2)
54. (<b, s, c> \mmember{} r)
55. e3 : E
56. d1 : \mBbbZ{}
57. (e3 < e2)
58. loc(e3) \mdownarrow{}\mmember{} accpts
59. header(e3) = ``pv11\_p1 p1a``
60. has-es-info-type(es;e3;f;Id \mtimes{} pv11\_p1\_Ballot\_Num())
61. d1 = 0
62. loc(e2) = (fst(msgval(e3)))
63. msg-authentic(info(e2)) = pv11\_p1\_init\_active()
64. l = loc(e3)
65. (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (snd(msgval(e3)))
66. <pv11\_p1\_upd\_bnum() msgval(e1) loc(e1), r> = pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid\000C}$;f;es;e3)
\mvdash{} \mdownarrow{}\mexists{}e':E
\mexists{}accepted:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
((e' < e)
\mwedge{} <bnum, accepted> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e')
\mwedge{} (<b, s, c> \mmember{} accepted))
By
Latex:
(D 0 THEN InstConcl [\mkleeneopen{}e3\mkleeneclose{};\mkleeneopen{}r\mkleeneclose{}]\mcdot{} THEN Auto THEN BLemma `pv11\_p1\_AcceptorState-classrel` THEN Auto)
Home
Index