Proof of Nuprl Lemma : pv11_p1

Step * 1 1 1 2 1 of Lemma pv11_p1_validity-lemma

.....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

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. header(e) = ``pv11_p1 p2a`` ∈ Name
21. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
22. ∀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'))))

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

34. (<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'))))))))

35. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()
36. e' c≤ e
⊢ pv11_p1_valid-proposal(Cmd;es;e';<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>;f)
BY
{ (D -3 THEN ExRepD THEN (FLemma `bag-member-sq-list-member` [-4] THENA Auto) THEN SqExRepD) }

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'))))

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

35. ¬(∃p2:Cmd. (<fst(snd(snd(msgval(e)))), p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e')))))
36. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()
37. e' c≤ e

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

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

2
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. ¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, fst(snd(snd(msgval(e)))), z8> ∈ snd(msgval(e')))))

37. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()
38. e' c≤ e
39. (<v2, fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ∈ snd(msgval(e')))
⊢ pv11_p1_valid-proposal(Cmd;es;e';<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>;f)

Latex:

Latex:
.....assertion..... 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. header(e) = ``pv11\_p1 p2a`` 21. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) 22. \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')))) 23. e' : E 24. delay : \mBbbZ{} 25. (e' < e) 26. loc(e') \mdownarrow{}\mmember{} ldrs 27. loc(e) \mdownarrow{}\mmember{} accpts 28. delay = 0 29. es-info-auth(es;e) = pv11\_p1\_init\_active() 30. (fst(msgval(e))) = loc(e') 31. header(e') = ``pv11\_p1 adopted`` 32. has-es-info-type(es;e';f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 33. (fst(msgval(e'))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e'))) 34. (<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')))))))) 35. (fst(snd(msgval(e)))) = (fst(msgval(e'))) 36. e' c\mleq{} e \mvdash{} pv11\_p1\_valid-proposal(Cmd;es;e';<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>f) By Latex: (D -3 THEN ExRepD THEN (FLemma `bag-member-sq-list-member` [-4] THENA Auto) THEN SqExRepD) Home Index