Proof of Nuprl Lemma : pv11_p1_decision_from_p2a

Step * 1 of Lemma pv11_p1_decision_from_p2a_acc

1. Cmd : {T:Type| valueall-type(T)}
2. f : pv11_p1_headers_type{i:l}(Cmd)
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))
13. accpts : bag(Id)
14. ldrs : bag(Id)
15. ldrs_uid : Id ─→ ℤ
16. reps : bag(Id)
17. e : E
18. s : ℤ
19. c : Cmd
20. a : Id
21. Inj(Id;ℤ;ldrs_uid)@i
22. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
23. pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e;s;c)@i'
24. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())@i
25. a ↓∈ accpts@i
⊢ ↓∃e':E
    ((e' < e)
    ∧ loc(e') ↓∈ accpts
    ∧ (s = (fst(snd(snd(msgval(e))))) ∈ ℤ)
    ∧ ((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())
    ∧ <loc(e), snd(snd(snd(msgval(e)))), s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'))
BY
{ (RepeatFor 2 (D (-3))
   THEN (RWO "make-Msg-as-mk-msg" (-3) THENA Auto)
   THEN (RewriteWithILF (-3) THENA Auto)
   THEN SquashExRepD
   THEN DVarSets
   THEN OnSomeHyp UseMessageConstraint
   THEN RepD
   THEN Reduce 0
   THEN (D 0 THEN InstConcl [⌈e1⌉]⋅ THEN Auto)) }

1
1. Cmd : Type
2. valueall-type(Cmd)
3. f : pv11_p1_headers_type{i:l}(Cmd)
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))
14. accpts : bag(Id)
15. ldrs : bag(Id)
16. ldrs_uid : Id ─→ ℤ
17. reps : bag(Id)
18. e : E
19. s : ℤ
20. c : Cmd
21. a : Id
22. Inj(Id;ℤ;ldrs_uid)@i
23. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
24. delay : ℤ@i'
25. client : Id@i'
26. loc(e) ↓∈ ldrs
27. e' : E
28. e' ≤loc e
29. header(e) = ``pv11_p1 p2b`` ∈ Name
30. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
31. delay = 0 ∈ ℤ
32. (fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num()
33. #(pv11_p1_CommanderStateFun(Cmd;accpts;f;fst(snd(msgval(e)));fst(snd(snd(msgval(e))));es.e';e)) < ...
34. client ↓∈ reps
35. ff = pv11_p1_init_active()
36. s = (fst(snd(snd(msgval(e))))) ∈ ℤ
37. (((header(e') = [propose] ∈ Name) ∧ has-es-info-type(es;e';f;ℤ × Cmd))
∧ ((↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))))
  ∧ (¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e'))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))))))
∧ ((fst(snd(msgval(e)))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())
∧ (<fst(snd(snd(msgval(e)))), c> = msgval(e') ∈ (ℤ × Cmd)))
∨ (((header(e') = ``pv11_p1 adopted`` ∈ Name)
   ∧ has-es-info-type(es;e';f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
  ∧ ((fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())
  ∧ ((<fst(snd(snd(msgval(e)))), c> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))
    ∧ (¬(∃p2:Cmd. (<fst(snd(snd(msgval(e)))), p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e')))))))
    ∨ (∃v2:pv11_p1_Ballot_Num()
        (<v2, fst(snd(snd(msgval(e)))), c> ↓∈ snd(msgval(e'))
        ∧ (¬(∃z5:pv11_p1_Ballot_Num()

              ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, fst(snd(snd(msgval(e)))), z8> ∈ snd(msgval(e')))))))))

∧ ((fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()))

38. (no ((pv11_p1_commander_output(Cmd;accpts;reps;f) <fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), c> o pv11_p1_p2b'b\000Case(Cmd;f)) o

        pv11_p1_CommanderState(Cmd;accpts;f) (fst(snd(msgval(e)))) (fst(snd(snd(msgval(e)))))) between e' and e)

39. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())@i
40. a ↓∈ accpts@i
41. ∀e:E
      ((header(e) ∈ pv11_p1_headers_no_inputs())
      ⇒ (↓∃e':E
            ∃delay:ℤ

             ((e' < e) ∧ make-msg-interface(delay;loc(e);info(e)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e'))))

42. e1 : E
43. d1 : ℤ
44. (e1 < e)
45. loc(e1) ↓∈ accpts
46. header(e1) = ``pv11_p1 p2a`` ∈ Name
47. has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
48. d1 = 0 ∈ ℤ
49. loc(e) = (fst(msgval(e1))) ∈ Id
50. es-info-auth(es;e) = pv11_p1_init_active()
51. (fst(msgval(e))) = loc(e1) ∈ Id
52. (fst(snd(msgval(e)))) = (fst(snd(msgval(e1)))) ∈ pv11_p1_Ballot_Num()
53. (fst(snd(snd(msgval(e))))) = (fst(snd(snd(msgval(e1))))) ∈ ℤ
54. (snd(snd(snd(msgval(e))))) = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e1))) ∈ pv11_p1_Ballot_Num()
55. (e1 < e)
56. loc(e1) ↓∈ accpts
57. s = (fst(snd(snd(msgval(e))))) ∈ ℤ
58. (fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num()
59. msgval(e) ∈ Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num()
⊢ <loc(e), snd(snd(snd(msgval(e)))), s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e1)

Latex:

Latex:
1. Cmd : \{T:Type| valueall-type(T)\} 2. f : pv11\_p1\_headers\_type\{i:l\}(Cmd) 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)) 13. accpts : bag(Id) 14. ldrs : bag(Id) 15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{} 16. reps : bag(Id) 17. e : E 18. s : \mBbbZ{} 19. c : Cmd 20. a : Id 21. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 22. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 23. pv11\_p1\_decision\{i:l\}(Cmd;f;accpts;ldrs;ldrs$_{uid}$;reps;es;e;s;c)@i' 24. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())@i 25. a \mdownarrow{}\mmember{} accpts@i \mvdash{} \mdownarrow{}\mexists{}e':E ((e' < e) \mwedge{} loc(e') \mdownarrow{}\mmember{} accpts \mwedge{} (s = (fst(snd(snd(msgval(e)))))) \mwedge{} ((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e)))))) \mwedge{} <loc(e), snd(snd(snd(msgval(e)))), s, c> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e')) By Latex: (RepeatFor 2 (D (-3)) THEN (RWO "make-Msg-as-mk-msg" (-3) THENA Auto) THEN (RewriteWithILF (-3) THENA Auto) THEN SquashExRepD THEN DVarSets THEN OnSomeHyp UseMessageConstraint THEN RepD THEN Reduce 0 THEN (D 0 THEN InstConcl [\mkleeneopen{}e1\mkleeneclose{}]\mcdot{} THEN Auto)) Home Index