Proof of Nuprl Lemma : pv11

Step * 1 3 1 of Lemma pv11_p1-validity

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. i : Id@i
20. p1 : ℤ@i
21. p2 : Cmd@i
22. loc(e) ↓∈ ldrs
23. e' : {e':E| e' ≤loc e }
24. header(e) = ``pv11_p1 p2b`` ∈ Name
25. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
26. 0 = 0 ∈ ℤ
27. (fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num()
28. #(pv11_p1_CommanderStateFun(Cmd;accpts;f;fst(snd(msgval(e)));fst(snd(snd(msgval(e))));es.e';e)) < ...
29. i ↓∈ reps
30. ff = pv11_p1_init_active()
31. p1 = (fst(snd(snd(msgval(e))))) ∈ ℤ
32. header(e') = ``pv11_p1 adopted`` ∈ Name
33. has-es-info-type(es;e';f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
34. (fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
35. b : pv11_p1_Ballot_Num()
36. <b, fst(snd(snd(msgval(e)))), p2> ↓∈ snd(msgval(e'))

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

38. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()
39. ∀e:E
      (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
      ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
      ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.  (start ≤loc e  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f)))
      ∧ ((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))
      ∧ ((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))
      ∧ ((header(e) = ``pv11_p1 p2a`` ∈ Name)
        ⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
        ⇒ pv11_p1_valid-p2a-message(Cmd;es;e;f)))
40. (<b, fst(snd(snd(msgval(e)))), p2> ∈ snd(msgval(e')))
41. (<b, p1, p2> ∈ snd(msgval(e')))
⊢ pv11_p1_valid-proposal(Cmd;es;e;<p1, p2>;f)
BY
{ ((Using [`e',⌈e'⌉] (BLemma `pv11_p1_valid-proposal-transitivity`)⋅
    THEN Auto
    THEN Try (Complete ((DSetVars  THEN Unhide THEN Auto))))
   THEN Assert ⌈pv11_p1_valid-adopted-message(Cmd;es;e';f)⌉⋅
   THEN Auto
   THEN RepeatFor 2 (UnfoldTopAb (-1))
   THEN (InstLemma `l_all_iff` [⌈parm{i'}⌉;⌈ℤ × Cmd⌉]⋅ THENA Auto)
   THEN (RWO "-1" (-2) THENA Auto)
   THEN Thin (-1)
   THEN BackThruHyp' (-1)
   THEN Auto) }

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. i : Id@i 20. p1 : \mBbbZ{}@i 21. p2 : Cmd@i 22. loc(e) \mdownarrow{}\mmember{} ldrs 23. e' : \{e':E| e' \mleq{}loc e \} 24. header(e) = ``pv11\_p1 p2b`` 25. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num()) 26. 0 = 0 27. (fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) 28. \#(pv11\_p1\_CommanderStateFun(Cmd;accpts;f;fst(snd(msgval(e)));fst(snd(snd(...)));es.e';e)) < ... 29. i \mdownarrow{}\mmember{} reps 30. ff = pv11\_p1\_init\_active() 31. p1 = (fst(snd(snd(msgval(e))))) 32. header(e') = ``pv11\_p1 adopted`` 33. has-es-info-type(es;e';f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 34. (fst(msgval(e'))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e'))) 35. b : pv11\_p1\_Ballot\_Num() 36. <b, fst(snd(snd(msgval(e)))), p2> \mdownarrow{}\mmember{} snd(msgval(e')) 37. \mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num() \mexists{}z8:Cmd. ((\muparrow{}(b < z5)) \mwedge{} (<z5, fst(snd(snd(msgval(e)))), z8> \mmember{} snd(msgval(e'))))) 38. (fst(snd(msgval(e)))) = (fst(msgval(e'))) 39. \mforall{}e:E (pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e;f) \mwedge{} pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e;f) \mwedge{} (\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))) \mwedge{} ((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)) \mwedge{} ((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)) \mwedge{} ((header(e) = ``pv11\_p1 p2a``) {}\mRightarrow{} has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) {}\mRightarrow{} pv11\_p1\_valid-p2a-message(Cmd;es;e;f))) 40. (<b, fst(snd(snd(msgval(e)))), p2> \mmember{} snd(msgval(e'))) 41. (<b, p1, p2> \mmember{} snd(msgval(e'))) \mvdash{} pv11\_p1\_valid-proposal(Cmd;es;e;<p1, p2>f) By Latex: ((Using [`e',\mkleeneopen{}e'\mkleeneclose{}] (BLemma `pv11\_p1\_valid-proposal-transitivity`)\mcdot{} THEN Auto THEN Try (Complete ((DSetVars THEN Unhide THEN Auto)))) THEN Assert \mkleeneopen{}pv11\_p1\_valid-adopted-message(Cmd;es;e';f)\mkleeneclose{}\mcdot{} THEN Auto THEN RepeatFor 2 (UnfoldTopAb (-1)) THEN (InstLemma `l\_all\_iff` [\mkleeneopen{}parm\{i'\}\mkleeneclose{};\mkleeneopen{}\mBbbZ{} \mtimes{} Cmd\mkleeneclose{}]\mcdot{} THENA Auto) THEN (RWO "-1" (-2) THENA Auto) THEN Thin (-1) THEN BackThruHyp' (-1) THEN Auto) Home Index