Proof of Nuprl Lemma : pv11_p1_unique

Step * 1 1 4 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. e5 : {e1:E| e1 ≤loc e' }
37. header(e5) = ``pv11_p1 preempted`` ∈ Name
38. has-es-info-type(es;e5;f;pv11_p1_Ballot_Num())
39. ↑(pv11_p1_is_bnum() msgval(e5))
40. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e5)) < msgval(e5))

41. (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 e')
42. header(e') = ``pv11_p1 p1b`` ∈ Name

43. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

44. (pv11_p1_upd_bnum() msgval(e5) loc(e5)) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
45. delay = 0 ∈ ℤ
46. loc(e1) = loc(e') ∈ Id
47. (pv11_p1_upd_bnum() msgval(e5) loc(e5)) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
48. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e5)

                                             loc(e5);es.e5;e'))) < pv11_p1_threshold(accpts)

49. msg-authentic(info(e1)) = pv11_p1_init_active()
50. bnum = (pv11_p1_upd_bnum() msgval(e5) loc(e5)) ∈ pv11_p1_Ballot_Num()
51. 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)
52. e3 : E
53. d1 : ℤ
54. (e3 < e2)
55. loc(e3) ↓∈ ldrs
56. e4 : {e':E| e' ≤loc e3 }
57. header(e4) = ``pv11_p1 preempted`` ∈ Name
58. has-es-info-type(es;e4;f;pv11_p1_Ballot_Num())
59. ↑(pv11_p1_is_bnum() msgval(e4))
60. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e4)) < msgval(e4))

61. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e4) loc(e4)) o pv11_p1_p1b'base(Cmd;f)) o

pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e4) loc(e4))) between e4 and e3)
62. header(e3) = ``pv11_p1 p1b`` ∈ Name

63. has-es-info-type(es;e3;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

64. (pv11_p1_upd_bnum() msgval(e4) loc(e4)) = (fst(snd(msgval(e3)))) ∈ pv11_p1_Ballot_Num()
65. d1 = 0 ∈ ℤ
66. loc(e2) = loc(e3) ∈ Id
67. (pv11_p1_upd_bnum() msgval(e4) loc(e4)) = (fst(snd(snd(msgval(e3))))) ∈ pv11_p1_Ballot_Num()
68. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e4)

                                             loc(e4);es.e4;e3))) < pv11_p1_threshold(accpts)

69. msg-authentic(info(e2)) = pv11_p1_init_active()
70. bnum = (pv11_p1_upd_bnum() msgval(e4) loc(e4)) ∈ pv11_p1_Ballot_Num()
71. 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)
⊢ pvals1 = pvals2 ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
BY

{ (SubstFor ⌈pvals1⌉ 0⋅ THEN HypSubst' (-1) 0 THEN DVarSets THEN UseLoclTri ⌈es⌉⌈e4⌉⌈e5⌉⋅) }

1
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. (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 e')
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(e4) loc(e4)) o pv11_p1_p1b'base(Cmd;f)) o

pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e4) loc(e4))) between e4 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 <loc e5)
⊢ (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e5) loc(e5);es.e5;e')))
= (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)

2
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

43. (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 e')
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))

                                             loc(e5);es.e5;e'))) < pv11_p1_threshold(accpts)

64. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e4) loc(e4)) o pv11_p1_p1b'base(Cmd;f)) o

pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e4) loc(e4))) between e4 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))

                                             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
⊢ (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e5) loc(e5);es.e5;e')))
= (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)

3
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

43. (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 e')
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))

                                             loc(e5);es.e5;e'))) < pv11_p1_threshold(accpts)

64. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e4) loc(e4)) o pv11_p1_p1b'base(Cmd;f)) o

pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e4) loc(e4))) between e4 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))

                                             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. (e5 <loc e4)
⊢ (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e5) loc(e5);es.e5;e')))
= (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)

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. e5 : \{e1:E| e1 \mleq{}loc e' \} 37. header(e5) = ``pv11\_p1 preempted`` 38. has-es-info-type(es;e5;f;pv11\_p1\_Ballot\_Num()) 39. \muparrow{}(pv11\_p1\_is\_bnum() msgval(e5)) 40. \muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e5)) < msgval(e5)) 41. (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 e') 42. header(e') = ``pv11\_p1 p1b`` 43. 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)) 44. (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5)) = (fst(snd(msgval(e')))) 45. delay = 0 46. loc(e1) = loc(e') 47. (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5)) = (fst(snd(snd(msgval(e'))))) 48. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e5) loc(e5);es.e5;e'))) < pv11\_p1\_threshold(accpts) 49. msg-authentic(info(e1)) = pv11\_p1\_init\_active() 50. bnum = (pv11\_p1\_upd\_bnum() msgval(e5) loc(e5)) 51. pvals1 = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e5) loc(e5);es.e5;e'))) 52. e3 : E 53. d1 : \mBbbZ{} 54. (e3 < e2) 55. loc(e3) \mdownarrow{}\mmember{} ldrs 56. e4 : \{e':E| e' \mleq{}loc e3 \} 57. header(e4) = ``pv11\_p1 preempted`` 58. has-es-info-type(es;e4;f;pv11\_p1\_Ballot\_Num()) 59. \muparrow{}(pv11\_p1\_is\_bnum() msgval(e4)) 60. \muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e4)) < msgval(e4)) 61. (no ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4)) o pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4))) between e4 and e3) 62. header(e3) = ``pv11\_p1 p1b`` 63. 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)) 64. (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4)) = (fst(snd(msgval(e3)))) 65. d1 = 0 66. loc(e2) = loc(e3) 67. (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4)) = (fst(snd(snd(msgval(e3))))) 68. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e4) loc(e4);es.e4;e3))) < pv11\_p1\_threshold(accpts) 69. msg-authentic(info(e2)) = pv11\_p1\_init\_active() 70. bnum = (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4)) 71. pvals2 = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e4) loc(e4);es.e4;e3))) \mvdash{} pvals1 = pvals2 By Latex: (SubstFor \mkleeneopen{}pvals1\mkleeneclose{} 0\mcdot{} THEN HypSubst' (-1) 0 THEN DVarSets THEN UseLoclTri \mkleeneopen{}es\mkleeneclose{}\mkleeneopen{}e4\mkleeneclose{}\mkleeneopen{}e5\mkleeneclose{}\mcdot{}) Home Index