Proof of Nuprl Lemma : pv11_p1_unique

Step * 1 1 4 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. (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. (e5 <loc e4)
76. let bnum1,active1,proposals1 = pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e5) in
let bnum2,active2,proposals2 = pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e4) in
case inr inr msgval(e5)
of inl(x) =>
True
| inr(x) =>
case x
  of inl(x) =>
  True
  | inr(bnum) =>
  (↑(bnum1  < bnum)) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (pv11_p1_upd_bnum() bnum loc(e5)) bnum2))
⊢ (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)
BY
{ (InstLemma `pv11_p1_ldr_mem_preempted` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈e5⌉;⌈e4⌉;⌈ldrs_uid⌉;

   ⌈pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e5)⌉;⌈pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e4)⌉;⌈inr inr msgval(e5)  ⌉]

   ⋅
   THENA (Auto THEN Try ((BLemma `pv11_p1_LeaderState-classrel` THEN Auto)))
   ) }

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

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)

⊢ inr inr msgval(e5)   ∈ pv11_p1_propose'base(Cmd;f) (+) pv11_p1_adopted'base(Cmd;f) (+) pv11_p1_preempted'base(Cmd;f)(

e5)

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. (e5 <loc e4)
76. let bnum1,active1,proposals1 = pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e5) in
let bnum2,active2,proposals2 = pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e4) in
case inr inr msgval(e5)
of inl(x) =>
True
| inr(x) =>
case x
  of inl(x) =>
  True
  | inr(bnum) =>
  (↑(bnum1  < bnum)) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (pv11_p1_upd_bnum() bnum loc(e5)) bnum2))
77. let bnum1,active1,proposals1 = pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e5) in
let bnum2,active2,proposals2 = pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e4) in
case inr inr msgval(e5)
of inl(x) =>
True
| inr(x) =>
case x
  of inl(x) =>
  True
  | inr(bnum) =>
  (↑(bnum1  < bnum)) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (pv11_p1_upd_bnum() bnum loc(e5)) bnum2))
⊢ (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 : 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. (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`` 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(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`` 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. (e5 <loc e4) 76. let bnum1,active1,proposals1 = pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;\000Ce5) in let bnum2,active2,proposals2 = pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e4) \000Cin case inr inr msgval(e5) of inl(x) => True | inr(x) => case x of inl(x) => True | inr(bnum) => (\muparrow{}(bnum1 < bnum)) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (pv11\_p1\_upd\_bnum() bn\000Cum loc(e5)) bnum2)) \mvdash{} (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))) By Latex: (InstLemma `pv11\_p1\_ldr\_mem\_preempted` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e5\mkleeneclose{};\mkleeneopen{}e4\mkleeneclose{};\mkleeneopen{}ldrs$_{uid}$\mkleeneclose{}; \mkleeneopen{}pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e5)\mkleeneclose{};\mkleeneopen{}pv11\_p1\_LeaderStateFun(Cmd;\000Cldrs$_{uid}$;f;es;e4)\mkleeneclose{}; \mkleeneopen{}inr inr msgval(e5) \mkleeneclose{}]\mcdot{} THENA (Auto THEN Try ((BLemma `pv11\_p1\_LeaderState-classrel` THEN Auto))) ) Home Index