Proof of Nuprl Lemma : pv11_p1

Step * 1 7 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)
⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f)
24. (header(e) = ``pv11_p1 p1b`` ∈ Name)
⇒

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

⇒ pv11_p1_valid-p1b-message(Cmd;es;e;f)
25. header(e) = ``pv11_p1 p2a`` ∈ Name@i
26. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i
27. ∀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'))))

28. e' : E
29. delay : ℤ
30. (e' < e)
31. loc(e') ↓∈ ldrs
32. loc(e) ↓∈ accpts
33. delay = 0 ∈ ℤ
34. es-info-auth(es;e) = pv11_p1_init_active()
35. (fst(msgval(e))) = loc(e') ∈ Id
36. header(e') = ``pv11_p1 adopted`` ∈ Name
37. has-es-info-type(es;e';f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
38. (fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()

39. (<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ↓∈ 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)))), snd(snd(snd(msgval(e))))> ↓∈ snd(msgval(e'))

    ∧ (¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, fst(snd(snd(msgval(e)))), z8> ∈ snd(msgval(e'))))))))

40. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()
⊢ pv11_p1_valid-proposal(Cmd;es;e;snd(snd(<fst(msgval(e))
                                          , fst(snd(msgval(e)))
                                          , fst(snd(snd(msgval(e))))
                                          , snd(snd(snd(msgval(e))))>));f)
BY
{ (D -2
   THEN SqExRepD
   THEN (FLemma `bag-member-sq-list-member` [-3] THENA Auto)
   THEN SqExRepD
   THEN (Assert ⌈e' c≤ e⌉ BY
               Auto)
   THEN (Assert pv11_p1_valid-proposal(Cmd;es;e';<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>;f)⋅
   THENM (FLemma `pv11_p1_valid-proposal-transitivity` [-1;-2] THEN Auto)
   )) }

1
.....assertion.....
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

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

39. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))

40. ¬(∃p2:Cmd. (<fst(snd(snd(msgval(e)))), p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e')))))
41. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()

42. (<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))))

43. e' c≤ e
⊢ pv11_p1_valid-proposal(Cmd;es;e';<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>;f)

2
.....assertion.....
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

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

41. ¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, fst(snd(snd(msgval(e)))), z8> ∈ snd(msgval(e')))))

42. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()
43. (<v2, fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ∈ snd(msgval(e')))
44. e' c≤ e
⊢ pv11_p1_valid-proposal(Cmd;es;e';<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>;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``) {}\mRightarrow{} has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) {}\mRightarrow{} pv11\_p1\_valid-adopted-message(Cmd;es;e;f) 24. (header(e) = ``pv11\_p1 p1b``) {}\mRightarrow{} 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)) {}\mRightarrow{} pv11\_p1\_valid-p1b-message(Cmd;es;e;f) 25. header(e) = ``pv11\_p1 p2a``@i 26. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)@i 27. \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')))) 28. e' : E 29. delay : \mBbbZ{} 30. (e' < e) 31. loc(e') \mdownarrow{}\mmember{} ldrs 32. loc(e) \mdownarrow{}\mmember{} accpts 33. delay = 0 34. es-info-auth(es;e) = pv11\_p1\_init\_active() 35. (fst(msgval(e))) = loc(e') 36. header(e') = ``pv11\_p1 adopted`` 37. has-es-info-type(es;e';f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 38. (fst(msgval(e'))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e'))) 39. (<fst(snd(snd(msgval(e)))) , snd(snd(snd(msgval(e)))) > \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e'))) \mwedge{} (\mneg{}(\mexists{}p2:Cmd. (<fst(snd(snd(msgval(e)))), p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) (\000Csnd(msgval(e'))))))) \mvee{} (\mexists{}v2:pv11\_p1\_Ballot\_Num() (<v2, fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> \mdownarrow{}\mmember{} snd(msgval(e')) \mwedge{} (\mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num() \mexists{}z8:Cmd. ((\muparrow{}(v2 < z5)) \mwedge{} (<z5, fst(snd(snd(msgval(e)))), z8> \mmember{} snd(msgval(e')))))))) 40. (fst(snd(msgval(e)))) = (fst(msgval(e'))) \mvdash{} pv11\_p1\_valid-proposal(Cmd;es;e;snd(snd(<fst(msgval(e)) , fst(snd(msgval(e))) , fst(snd(snd(msgval(e)))) , snd(snd(snd(msgval(e))))>));f) By Latex: (D -2 THEN SqExRepD THEN (FLemma `bag-member-sq-list-member` [-3] THENA Auto) THEN SqExRepD THEN (Assert \mkleeneopen{}e' c\mleq{} e\mkleeneclose{} BY Auto) THEN (Assert pv11\_p1\_valid-proposal(Cmd;es;e';<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>\000Cf)\mcdot{} THENM (FLemma `pv11\_p1\_valid-proposal-transitivity` [-1;-2] THEN Auto) )) Home Index