Proof of Nuprl Lemma : pv11_p1_adopted_from

Step * 1 of Lemma pv11_p1_adopted_from_acceptor

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. e : E@i
14. accpts : bag(Id)@i
15. ldrs : bag(Id)@i
16. ldrs_uid : Id ─→ ℤ@i
17. reps : bag(Id)@i
18. bnum : pv11_p1_Ballot_Num()@i
19. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
20. b : pv11_p1_Ballot_Num()@i
21. s : ℤ@i
22. c : Cmd@i
23. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
24. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)@i
25. (<b, s, c> ∈ pvals)@i
26. e' : E
27. delay : ℤ
28. (e' < e)
29. loc(e') ↓∈ ldrs
30. (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')
31. header(e') = ``pv11_p1 p1b`` ∈ Name

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

     ((e' < e) ∧ <bnum, accepted> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e') ∧ (<b, s, c> ∈ accepted))

BY
{ ((InstLemma `pv11_p1_scout_from_acc` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈e'⌉;⌈accpts⌉;

    ⌈pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')⌉;⌈pv11_p1_init_ballot_num() loc(e')⌉]⋅

    THENA (Auto THEN BLemma `pv11_p1_ScoutState-classrel` THEN Auto)
    )
   THEN UsePairEta [1] (-1)
   THEN (RWO "-2<" (-1) THENA Auto)
   THEN (InstHyp [⌈<b, s, c>⌉] (-1)⋅ THENA Auto)
   THEN ExRepD
   THEN RenameVar `e1' (-6)
   THEN (UseMessageConstraint (-2) THENA Auto)
   THEN RepD) }

1
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

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

33. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
34. delay = 0 ∈ ℤ
35. loc(e) = loc(e') ∈ Id
36. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
37. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
38. msg-authentic(info(e)) = pv11_p1_init_active()
39. bnum = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num()
40. pvals
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
41. ∀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_init_ballot_num() loc(e'), pv11_p1_init_ballot_num() loc(e'), r> ∈
                pv11_p1_p1b'base(Cmd;f)(e'@0)
             ∧ (p ∈ r))))
42. e1 : E
43. l : Id
44. r : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
45. e1 ≤loc e'
46. <l, pv11_p1_init_ballot_num() loc(e'), pv11_p1_init_ballot_num() loc(e'), r> ∈ pv11_p1_p1b'base(Cmd;f)(e1)
47. (<b, s, c> ∈ r)
48. e'@0 : E
49. d1 : ℤ
50. (e'@0 < e1)
51. loc(e'@0) ↓∈ accpts
52. header(e'@0) = ``pv11_p1 p1a`` ∈ Name
53. has-es-info-type(es;e'@0;f;Id × pv11_p1_Ballot_Num())
54. d1 = 0 ∈ ℤ
55. loc(e1) = (fst(msgval(e'@0))) ∈ Id
56. msg-authentic(info(e1)) = pv11_p1_init_active()
57. l = loc(e'@0) ∈ Id
58. (pv11_p1_init_ballot_num() loc(e')) = (snd(msgval(e'@0))) ∈ pv11_p1_Ballot_Num()
59. <pv11_p1_init_ballot_num() loc(e'), r>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0)
∈ (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))

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. e : E@i 14. accpts : bag(Id)@i 15. ldrs : bag(Id)@i 16. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 17. reps : bag(Id)@i 18. bnum : pv11\_p1\_Ballot\_Num()@i 19. pvals : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 20. b : pv11\_p1\_Ballot\_Num()@i 21. s : \mBbbZ{}@i 22. c : Cmd@i 23. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 24. <bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e)@i 25. (<b, s, c> \mmember{} pvals)@i 26. e' : E 27. delay : \mBbbZ{} 28. (e' < e) 29. loc(e') \mdownarrow{}\mmember{} ldrs 30. (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') 31. header(e') = ``pv11\_p1 p1b`` 32. 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)) 33. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(msgval(e')))) 34. delay = 0 35. loc(e) = loc(e') 36. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(snd(msgval(e'))))) 37. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e'))) < pv11\_p1\_threshold(accpts) 38. msg-authentic(info(e)) = pv11\_p1\_init\_active() 39. bnum = (pv11\_p1\_init\_ballot\_num() loc(e')) 40. pvals = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e'))) \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: ((InstLemma `pv11\_p1\_scout\_from\_acc` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e'\mkleeneclose{};\mkleeneopen{}accpts\mkleeneclose{}; \mkleeneopen{}pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e')\mkleeneclose{}; \mkleeneopen{}pv11\_p1\_init\_ballot\_num() loc(e')\mkleeneclose{}]\mcdot{} THENA (Auto THEN BLemma `pv11\_p1\_ScoutState-classrel` THEN Auto) ) THEN UsePairEta [1] (-1) THEN (RWO "-2<" (-1) THENA Auto) THEN (InstHyp [\mkleeneopen{}<b, s, c>\mkleeneclose{}] (-1)\mcdot{} THENA Auto) THEN ExRepD THEN RenameVar `e1' (-6) THEN (UseMessageConstraint (-2) THENA Auto) THEN RepD) Home Index