Step
*
1
1
1
1
3
of Lemma
pv11_p1_unique_adopted
1. Cmd : {T:Type| valueall-type(T)} @i'
2. f : pv11_p1_headers_type{i:l}(Cmd)@i'
3. (f [decision]) = (ℤ × Cmd) ∈ Type
4. (f [propose]) = (ℤ × Cmd) ∈ Type
5. (f ``pv11_p1 adopted``) = (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)) ∈ Type
6. (f ``pv11_p1 preempted``) = pv11_p1_Ballot_Num() ∈ Type
7. (f ``pv11_p1 p2b``) = (Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num()) ∈ Type
8. (f ``pv11_p1 p2a``) = (Id × pv11_p1_Ballot_Num() × ℤ × Cmd) ∈ Type
9. (f ``pv11_p1 p1b``)
= (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
∈ Type
10. (f ``pv11_p1 p1a``) = (Id × pv11_p1_Ballot_Num()) ∈ Type
11. f ∈ Name ─→ Type
12. es : EO+(Message(f))@i'
13. e1 : E@i
14. e2 : 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. pvals1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
21. pvals2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
22. proposals1 : (ℤ × Cmd) List@i
23. proposals2 : (ℤ × Cmd) List@i
24. b1 : 𝔹@i
25. b2 : 𝔹@i
26. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
27. <bnum, pvals1> ∈ pv11_p1_adopted'base(Cmd;f)(e1)@i
28. <bnum, pvals2> ∈ pv11_p1_adopted'base(Cmd;f)(e2)@i
29. <bnum, b1, proposals1> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)@i
30. <bnum, b2, proposals2> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e2)@i
31. (e1 <loc e2)@i
32. e' : E
33. delay : ℤ
34. (e' < e1)
35. loc(e') ↓∈ ldrs
36. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e')) o pv11_p1_p1b'base(Cmd;f)) o
pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e'))) prior to e')
37. header(e') = ``pv11_p1 p1b`` ∈ Name
38. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
39. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
40. delay = 0 ∈ ℤ
41. loc(e1) = loc(e') ∈ Id
42. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
43. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
44. msg-authentic(info(e1)) = pv11_p1_init_active()
45. bnum = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num()
46. pvals1
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
47. e3 : E
48. d1 : ℤ
49. (e3 < e2)
50. loc(e3) ↓∈ ldrs
51. ∀e':E
((e' <loc e3)
⇒ (∀w:Interface
(¬w ∈ ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e3)) o pv11_p1_p1b'base(Cmd;f)) o
pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e3)))(e'))))
52. header(e3) = ``pv11_p1 p1b`` ∈ Name
53. has-es-info-type(es;e3;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
54. (pv11_p1_init_ballot_num() loc(e3)) = (fst(snd(msgval(e3)))) ∈ pv11_p1_Ballot_Num()
55. d1 = 0 ∈ ℤ
56. loc(e2) = loc(e3) ∈ Id
57. (pv11_p1_init_ballot_num() loc(e3)) = (fst(snd(snd(msgval(e3))))) ∈ pv11_p1_Ballot_Num()
58. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e3);es;e3))) < pv11_p1_threshold(accpts)
59. msg-authentic(info(e2)) = pv11_p1_init_active()
60. bnum = (pv11_p1_init_ballot_num() loc(e3)) ∈ pv11_p1_Ballot_Num()
61. pvals2
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e3);es;e3)))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
62. (e' <loc e3)
63. pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e3)) loc(e') msgval(e') ∈
(pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e3)) o pv11_p1_p1b'base(Cmd;f))(e')
64. pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e') ∈ pv11_p1_ScoutState(Cmd;accpts;f)
(pv11_p1_init_ballot_num() loc(e3))(
e')
65. msgval(e') ∈ Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
⊢ pv11_p1_adopted'send(Cmd;f) loc(e1) <bnum, pvals1> ↓∈ pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_init_ballot_num() lo\000Cc(e3)) loc(e') msgval(e')
pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')
BY
{ (Repeat (BagMemberD 0) THEN Auto) }
Latex:
Latex:
1. Cmd : \{T:Type| valueall-type(T)\} @i'
2. f : pv11\_p1\_headers\_type\{i:l\}(Cmd)@i'
3. (f [decision]) = (\mBbbZ{} \mtimes{} Cmd)
4. (f [propose]) = (\mBbbZ{} \mtimes{} Cmd)
5. (f ``pv11\_p1 adopted``) = (pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
6. (f ``pv11\_p1 preempted``) = pv11\_p1\_Ballot\_Num()
7. (f ``pv11\_p1 p2b``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
8. (f ``pv11\_p1 p2a``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
9. (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))
10. (f ``pv11\_p1 p1a``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num())
11. f \mmember{} Name {}\mrightarrow{} Type
12. es : EO+(Message(f))@i'
13. e1 : E@i
14. e2 : 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. pvals1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
21. pvals2 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
22. proposals1 : (\mBbbZ{} \mtimes{} Cmd) List@i
23. proposals2 : (\mBbbZ{} \mtimes{} Cmd) List@i
24. b1 : \mBbbB{}@i
25. b2 : \mBbbB{}@i
26. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
27. <bnum, pvals1> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e1)@i
28. <bnum, pvals2> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e2)@i
29. <bnum, b1, proposals1> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e1)@i
30. <bnum, b2, proposals2> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e2)@i
31. (e1 <loc e2)@i
32. e' : E
33. delay : \mBbbZ{}
34. (e' < e1)
35. loc(e') \mdownarrow{}\mmember{} ldrs
36. (no ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_init\_ballot\_num() loc(e')) o
pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f)
(pv11\_p1\_init\_ballot\_num() loc(e'))) prior to e')
37. header(e') = ``pv11\_p1 p1b``
38. 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))
39. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(msgval(e'))))
40. delay = 0
41. loc(e1) = loc(e')
42. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(snd(msgval(e')))))
43. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num()
loc(e');es;e'))) < pv11\_p1\_threshold(accpts)
44. msg-authentic(info(e1)) = pv11\_p1\_init\_active()
45. bnum = (pv11\_p1\_init\_ballot\_num() loc(e'))
46. pvals1 = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e')))
47. e3 : E
48. d1 : \mBbbZ{}
49. (e3 < e2)
50. loc(e3) \mdownarrow{}\mmember{} ldrs
51. \mforall{}e':E
((e' <loc e3)
{}\mRightarrow{} (\mforall{}w:Interface
(\mneg{}w \mmember{} ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_init\_ballot\_num() loc(e3)) o
pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f)
(pv11\_p1\_init\_ballot\_num() loc(e3)))(e'))))
52. header(e3) = ``pv11\_p1 p1b``
53. 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))
54. (pv11\_p1\_init\_ballot\_num() loc(e3)) = (fst(snd(msgval(e3))))
55. d1 = 0
56. loc(e2) = loc(e3)
57. (pv11\_p1\_init\_ballot\_num() loc(e3)) = (fst(snd(snd(msgval(e3)))))
58. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num()
loc(e3);es;e3))) < pv11\_p1\_threshold(accpts)
59. msg-authentic(info(e2)) = pv11\_p1\_init\_active()
60. bnum = (pv11\_p1\_init\_ballot\_num() loc(e3))
61. pvals2 = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e3);es;e3)))
62. (e' <loc e3)
63. pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_init\_ballot\_num() loc(e3)) loc(e') msgval(e') \mmember{}
(pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_init\_ballot\_num() loc(e3)) o
pv11\_p1\_p1b'base(Cmd;f))(e')
64. pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e') \mmember{}
pv11\_p1\_ScoutState(Cmd;accpts;f) (pv11\_p1\_init\_ballot\_num() loc(e3))(e')
65. msgval(e') \mmember{} Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)
\mvdash{} pv11\_p1\_adopted'send(Cmd;f) loc(e1) <bnum, pvals1> \mdownarrow{}\mmember{} pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_\000Cinit\_ballot\_num() loc(e3)) loc(e')
msgval(e')
pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e')
By
Latex:
(Repeat (BagMemberD 0) THEN Auto)
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