Proof of Nuprl Lemma : pv11_p1

Step * 1 4 1 1 1 of Lemma pv11_p1_validity-lemma

1. Cmd : {T:Type| valueall-type(T)}
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
      ((e1 < e)
      ⇒ (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e1;f)
         ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e1;f)
         ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.
              (start ≤loc e1  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
         ∧ ((header(e1) = ``pv11_p1 adopted`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-adopted-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p1b`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id
              × pv11_p1_Ballot_Num()
              × pv11_p1_Ballot_Num()
              × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-p1b-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p2a`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
           ⇒ pv11_p1_valid-p2a-message(Cmd;es;e1;f))))
20. pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
21. pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
22. ∀b:pv11_p1_Ballot_Num(). ∀start:E.  (start ≤loc e  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f))
23. header(e) = ``pv11_p1 adopted`` ∈ Name@i
24. has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
25. p1 : ℤ@i
26. p2 : Cmd@i
27. (<p1, p2> ∈ map(λp.(snd(p));snd(msgval(e))))@i
28. ∀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'))))

29. e' : E
30. delay : ℤ
31. (e' < e)
32. loc(e') ↓∈ ldrs
33. (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')
34. header(e') = ``pv11_p1 p1b`` ∈ Name

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

36. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
37. delay = 0 ∈ ℤ
38. loc(e) = loc(e') ∈ Id
39. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
40. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
41. es-info-auth(es;e) = pv11_p1_init_active()
42. (fst(msgval(e))) = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num()
43. (snd(msgval(e)))
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
44. pv11_p1_valid-ScoutState(Cmd;accpts;pv11_p1_init_ballot_num() loc(e');es-init(es;e');es;e';f)
45. es-init(es;e') ≤loc e'
⊢ ↓pv11_p1_valid-proposal(Cmd;es;e;<p1, p2>;f)
BY
{ PromoteHyp (-1) (-12) }

1
1. Cmd : {T:Type| valueall-type(T)}
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

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

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

37. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
38. delay = 0 ∈ ℤ
39. loc(e) = loc(e') ∈ Id
40. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
41. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
42. es-info-auth(es;e) = pv11_p1_init_active()
43. (fst(msgval(e))) = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num()
44. (snd(msgval(e)))
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
45. pv11_p1_valid-ScoutState(Cmd;accpts;pv11_p1_init_ballot_num() loc(e');es-init(es;e');es;e';f)
⊢ ↓pv11_p1_valid-proposal(Cmd;es;e;<p1, p2>;f)

Latex:

Latex:
1. Cmd : \{T:Type| valueall-type(T)\} 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. accpts : bag(Id)@i 14. ldrs : bag(Id)@i 15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 16. reps : bag(Id)@i 17. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 18. e : E@i 19. \mforall{}e1:E ((e1 < e) {}\mRightarrow{} (pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e1;f) \mwedge{} pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e1;f) \mwedge{} (\mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}start:E. (start \mleq{}loc e1 {}\mRightarrow{} pv11\_p1\_valid-ScoutState(Cmd;accpts;b;start;es;e1;f))) \mwedge{} ((header(e1) = ``pv11\_p1 adopted``) {}\mRightarrow{} has-es-info-type(es;e1;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) {}\mRightarrow{} pv11\_p1\_valid-adopted-message(Cmd;es;e1;f)) \mwedge{} ((header(e1) = ``pv11\_p1 p1b``) {}\mRightarrow{} has-es-info-type(es;e1;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) {}\mRightarrow{} pv11\_p1\_valid-p1b-message(Cmd;es;e1;f)) \mwedge{} ((header(e1) = ``pv11\_p1 p2a``) {}\mRightarrow{} has-es-info-type(es;e1;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) {}\mRightarrow{} pv11\_p1\_valid-p2a-message(Cmd;es;e1;f)))) 20. pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e;f) 21. pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e;f) 22. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}start:E. (start \mleq{}loc e {}\mRightarrow{} pv11\_p1\_valid-ScoutState(Cmd;accpts;b;start;es;e;f)) 23. header(e) = ``pv11\_p1 adopted``@i 24. has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))@i 25. p1 : \mBbbZ{}@i 26. p2 : Cmd@i 27. (<p1, p2> \mmember{} map(\mlambda{}p.(snd(p));snd(msgval(e))))@i 28. \mforall{}e:E ((header(e) \mmember{} pv11\_p1\_headers\_no\_inputs()) {}\mRightarrow{} (\mdownarrow{}\mexists{}e':E \mexists{}delay:\mBbbZ{} ((e' < e) \mwedge{} make-msg-interface(delay;loc(e);info(e)) \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}$;reps;f)(e')))) 29. e' : E 30. delay : \mBbbZ{} 31. (e' < e) 32. loc(e') \mdownarrow{}\mmember{} ldrs 33. (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') 34. header(e') = ``pv11\_p1 p1b`` 35. 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)) 36. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(msgval(e')))) 37. delay = 0 38. loc(e) = loc(e') 39. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(snd(msgval(e'))))) 40. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e'))) < pv11\_p1\_threshold(accpts) 41. es-info-auth(es;e) = pv11\_p1\_init\_active() 42. (fst(msgval(e))) = (pv11\_p1\_init\_ballot\_num() loc(e')) 43. (snd(msgval(e))) = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e'))) 44. pv11\_p1\_valid-ScoutState(Cmd;accpts;pv11\_p1\_init\_ballot\_num() loc(e');es-init(es;e');es;e';f) 45. es-init(es;e') \mleq{}loc e' \mvdash{} \mdownarrow{}pv11\_p1\_valid-proposal(Cmd;es;e;<p1, p2>f) By Latex: PromoteHyp (-1) (-12) Home Index