Step
*
1
1
4
2
3
2
of Lemma
pv11_p1_unique_adopted
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. e1 : E@i
15. e2 : E@i
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ─→ ℤ@i
19. reps : bag(Id)@i
20. bnum : pv11_p1_Ballot_Num()@i
21. pvals1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
22. pvals2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
23. proposals1 : (ℤ × Cmd) List@i
24. proposals2 : (ℤ × Cmd) List@i
25. b1 : 𝔹@i
26. b2 : 𝔹@i
27. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
28. <bnum, pvals1> ∈ pv11_p1_adopted'base(Cmd;f)(e1)@i
29. <bnum, pvals2> ∈ pv11_p1_adopted'base(Cmd;f)(e2)@i
30. <bnum, b1, proposals1> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)@i
31. <bnum, b2, proposals2> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e2)@i
32. (e1 <loc e2)@i
33. e' : E
34. delay : ℤ
35. (e' < e1)
36. loc(e') ↓∈ ldrs
37. e5 : E
38. e5 ≤loc e'
39. header(e5) = ``pv11_p1 preempted`` ∈ Name
40. has-es-info-type(es;e5;f;pv11_p1_Ballot_Num())
41. ↑(pv11_p1_is_bnum() msgval(e5))
42. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e5)) < msgval(e5))
43. ∀e'@0:E
((e'@0 <loc e')
⇒ e5 ≤loc e'@0
⇒ (∀w:Interface
(¬w ∈ ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e5) loc(e5)) o
pv11_p1_p1b'base(Cmd;f)) o pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e5) loc(e5)))(
e'@0))))
44. header(e') = ``pv11_p1 p1b`` ∈ Name
45. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
46. (pv11_p1_upd_bnum() msgval(e5) loc(e5)) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
47. delay = 0 ∈ ℤ
48. loc(e1) = loc(e') ∈ Id
49. (pv11_p1_upd_bnum() msgval(e5) loc(e5)) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
50. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e5)
loc(e5);es.e5;e'))) < pv11_p1_threshold(accpts)
51. msg-authentic(info(e1)) = pv11_p1_init_active()
52. bnum = (pv11_p1_upd_bnum() msgval(e5) loc(e5)) ∈ pv11_p1_Ballot_Num()
53. pvals1
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e5) loc(e5);es.e5;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
54. e3 : E
55. d1 : ℤ
56. (e3 < e2)
57. loc(e3) ↓∈ ldrs
58. e4 : E
59. e4 ≤loc e3
60. header(e4) = ``pv11_p1 preempted`` ∈ Name
61. has-es-info-type(es;e4;f;pv11_p1_Ballot_Num())
62. ↑(pv11_p1_is_bnum() msgval(e4))
63. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e4)) < msgval(e4))
64. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e5) loc(e5)) o pv11_p1_p1b'base(Cmd;f)) o
pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e5) loc(e5))) between e5 and e3)
65. header(e3) = ``pv11_p1 p1b`` ∈ Name
66. has-es-info-type(es;e3;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
67. (pv11_p1_upd_bnum() msgval(e4) loc(e4)) = (fst(snd(msgval(e3)))) ∈ pv11_p1_Ballot_Num()
68. d1 = 0 ∈ ℤ
69. loc(e2) = loc(e3) ∈ Id
70. (pv11_p1_upd_bnum() msgval(e4) loc(e4)) = (fst(snd(snd(msgval(e3))))) ∈ pv11_p1_Ballot_Num()
71. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e4)
loc(e4);es.e4;e3))) < pv11_p1_threshold(accpts)
72. msg-authentic(info(e2)) = pv11_p1_init_active()
73. bnum = (pv11_p1_upd_bnum() msgval(e4) loc(e4)) ∈ pv11_p1_Ballot_Num()
74. pvals2
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e4) loc(e4);es.e4;e3)))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
75. e4 = e5 ∈ E
76. (e3 <loc e')
77. pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e5) loc(e5)) loc(e3) msgval(e3) ∈
(pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e5) loc(e5)) o pv11_p1_p1b'base(Cmd;f))(e3)
78. msgval(e5) ∈ pv11_p1_Ballot_Num()
⊢ pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e5) loc(e5);es.e5;e3) ∈ pv11_p1_ScoutState(Cmd;accpts;f)
(pv11_p1_upd_bnum() msgval(e5)
loc(e5))(e3)
BY
{ (BLemma `pv11_p1_ScoutState-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. e1 : E@i
15. e2 : E@i
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
19. reps : bag(Id)@i
20. bnum : pv11\_p1\_Ballot\_Num()@i
21. pvals1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
22. pvals2 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
23. proposals1 : (\mBbbZ{} \mtimes{} Cmd) List@i
24. proposals2 : (\mBbbZ{} \mtimes{} Cmd) List@i
25. b1 : \mBbbB{}@i
26. b2 : \mBbbB{}@i
27. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
28. <bnum, pvals1> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e1)@i
29. <bnum, pvals2> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e2)@i
30. <bnum, b1, proposals1> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e1)@i
31. <bnum, b2, proposals2> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e2)@i
32. (e1 <loc e2)@i
33. e' : E
34. delay : \mBbbZ{}
35. (e' < e1)
36. loc(e') \mdownarrow{}\mmember{} ldrs
37. e5 : E
38. e5 \mleq{}loc e'
39. header(e5) = ``pv11\_p1 preempted``
40. has-es-info-type(es;e5;f;pv11\_p1\_Ballot\_Num())
41. \muparrow{}(pv11\_p1\_is\_bnum() msgval(e5))
42. \muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e5)) < msgval(e5))
43. \mforall{}e'@0:E
((e'@0 <loc e')
{}\mRightarrow{} e5 \mleq{}loc e'@0
{}\mRightarrow{} (\mforall{}w:Interface
(\mneg{}w \mmember{} ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5)) o
pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f)
(pv11\_p1\_upd\_bnum() msgval(e5) loc(e5)))(e'@0))))
44. header(e') = ``pv11\_p1 p1b``
45. 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))
46. (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5)) = (fst(snd(msgval(e'))))
47. delay = 0
48. loc(e1) = loc(e')
49. (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5)) = (fst(snd(snd(msgval(e')))))
50. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e5)
loc(e5);es.e5;e'))) < pv11\_p1\_threshold(accpts)
51. msg-authentic(info(e1)) = pv11\_p1\_init\_active()
52. bnum = (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5))
53. pvals1
= (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e5) loc(e5);es.e5;e')))
54. e3 : E
55. d1 : \mBbbZ{}
56. (e3 < e2)
57. loc(e3) \mdownarrow{}\mmember{} ldrs
58. e4 : E
59. e4 \mleq{}loc e3
60. header(e4) = ``pv11\_p1 preempted``
61. has-es-info-type(es;e4;f;pv11\_p1\_Ballot\_Num())
62. \muparrow{}(pv11\_p1\_is\_bnum() msgval(e4))
63. \muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e4)) < msgval(e4))
64. (no ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5)) o
pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f)
(pv11\_p1\_upd\_bnum() msgval(e5) loc(e5))) between e5 and e3)
65. header(e3) = ``pv11\_p1 p1b``
66. has-es-info-type(es;e3;f;Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
67. (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4)) = (fst(snd(msgval(e3))))
68. d1 = 0
69. loc(e2) = loc(e3)
70. (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4)) = (fst(snd(snd(msgval(e3)))))
71. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e4)
loc(e4);es.e4;e3))) < pv11\_p1\_threshold(accpts)
72. msg-authentic(info(e2)) = pv11\_p1\_init\_active()
73. bnum = (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4))
74. pvals2
= (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e4) loc(e4);es.e4;e3)))
75. e4 = e5
76. (e3 <loc e')
77. pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5)) loc(e3) msgval(e3) \mmember{}
(pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5)) o
pv11\_p1\_p1b'base(Cmd;f))(e3)
78. msgval(e5) \mmember{} pv11\_p1\_Ballot\_Num()
\mvdash{} pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e5) loc(e5);es.e5;e3) \mmember{}
pv11\_p1\_ScoutState(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5))(e3)
By
Latex:
(BLemma `pv11\_p1\_ScoutState-classrel` THEN Auto)
Home
Index