Proof of Nuprl Lemma : pv11_p1_pvalue_from

Step * 1 4 1 of Lemma pv11_p1_pvalue_from_p2a

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. 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)
      ⇒ (∀p:pv11_p1_Ballot_Num() × ℤ × Cmd
            (pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e1;p;f)
            ⇒ (↓∃e':E. (e' c≤ e1 ∧ pv11_p1_from-p2a{i:l}(Cmd;es;e';p))))))
20. p1 : pv11_p1_Ballot_Num()@i
21. p3 : ℤ@i
22. p4 : Cmd@i
23. header(e) = ``pv11_p1 adopted`` ∈ Name@i'
24. (<p1, p3, p4> ∈ snd(msgval(e)))@i'
25. has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
26. ∀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'))))

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

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

{ ((Assert (<p1, p3, p4> ∈ snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) BY

          Auto)⋅
   THEN (FLemma `pv11_p1_scout_fun_from_acc` [-1] THENA Auto)
   THEN ExRepD
   THEN D -3) }

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

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

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

34. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
35. delay = 0 ∈ ℤ
36. loc(e) = loc(e') ∈ Id
37. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
38. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
39. es-info-auth(es;e) = pv11_p1_init_active()
40. (fst(msgval(e))) = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num()
41. (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)
42. (<p1, p3, p4> ∈ snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
43. e1 : E
44. l : Id
45. r : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
46. (e1 <loc e')
47. <l, pv11_p1_init_ballot_num() loc(e'), pv11_p1_init_ballot_num() loc(e'), r> ∈ pv11_p1_p1b'base(Cmd;f)(e1)
48. (<p1, p3, p4> ∈ r)
⊢ ↓∃e':E. (e' c≤ e ∧ pv11_p1_from-p2a{i:l}(Cmd;es;e';<p1, p3, p4>))

2
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

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

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

34. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
35. delay = 0 ∈ ℤ
36. loc(e) = loc(e') ∈ Id
37. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
38. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
39. es-info-auth(es;e) = pv11_p1_init_active()
40. (fst(msgval(e))) = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num()
41. (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)
42. (<p1, p3, p4> ∈ snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
43. e1 : E
44. l : Id
45. r : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
46. e1 = e' ∈ E
47. <l, pv11_p1_init_ballot_num() loc(e'), pv11_p1_init_ballot_num() loc(e'), r> ∈ pv11_p1_p1b'base(Cmd;f)(e1)
48. (<p1, p3, p4> ∈ r)
⊢ ↓∃e':E. (e' c≤ e ∧ pv11_p1_from-p2a{i:l}(Cmd;es;e';<p1, p3, p4>))

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. 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{} (\mforall{}p:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd (pv11\_p1\_pvalue\{i:l\}(Cmd;ldrs$_{uid}$;accpts;es;e1;p;f) {}\mRightarrow{} (\mdownarrow{}\mexists{}e':E. (e' c\mleq{} e1 \mwedge{} pv11\_p1\_from-p2a\{i:l\}(Cmd;es;e';p)))))) 20. p1 : pv11\_p1\_Ballot\_Num()@i 21. p3 : \mBbbZ{}@i 22. p4 : Cmd@i 23. header(e) = ``pv11\_p1 adopted``@i' 24. (<p1, p3, p4> \mmember{} snd(msgval(e)))@i' 25. has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 26. \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')))) 27. e' : E 28. delay : \mBbbZ{} 29. (e' < e) 30. loc(e') \mdownarrow{}\mmember{} ldrs 31. (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') 32. header(e') = ``pv11\_p1 p1b`` 33. 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)) 34. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(msgval(e')))) 35. delay = 0 36. loc(e) = loc(e') 37. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(snd(msgval(e'))))) 38. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e'))) < pv11\_p1\_threshold(accpts) 39. es-info-auth(es;e) = pv11\_p1\_init\_active() 40. (fst(msgval(e))) = (pv11\_p1\_init\_ballot\_num() loc(e')) 41. (snd(msgval(e))) = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e'))) \mvdash{} \mdownarrow{}\mexists{}e':E. (e' c\mleq{} e \mwedge{} pv11\_p1\_from-p2a\{i:l\}(Cmd;es;e';<p1, p3, p4>)) By Latex: ((Assert (<p1, p3, p4> \mmember{} snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e'))) BY Auto)\mcdot{} THEN (FLemma `pv11\_p1\_scout\_fun\_from\_acc` [-1] THENA Auto) THEN ExRepD THEN D -3) Home Index