Proof of Nuprl Lemma : pv11_p1_consistency

Step * 1 2 1 1 2 of Lemma pv11_p1_consistency_lemma

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. Inj(Id;ℤ;ldrs_uid)@i
19. e : E@i
20. ∀e1:E
      ((e1 < e)
      ⇒ (∀b:pv11_p1_Ballot_Num(). ∀k:ℤ. ∀v:Cmd.
            (pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e1;<b, k, v>;f)
            ⇒ (∀e':E. ∀w:Cmd.
                  (pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e';k;w)
                  ⇒ has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
                  ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e'))))) b))
                  ⇒ (w = v ∈ Cmd))))))
21. b : pv11_p1_Ballot_Num()@i
22. k : ℤ@i
23. v : Cmd@i
24. pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e;<b, k, v>;f)@i'
25. header(e) = ``pv11_p1 p2a`` ∈ Name
26. <b, k, v> = (snd(msgval(e))) ∈ (pv11_p1_Ballot_Num() × ℤ × Cmd)
27. e' : E@i
28. w : Cmd@i
29. pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e';k;w)@i'
30. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())@i
31. ↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e'))))) b)@i
32. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
33. ∀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'))))

34. e1 : E
35. delay : ℤ
36. (e1 < e)
37. pv11_p1_p2a'send(Cmd;f) loc(e)
    <fst(msgval(e)), fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ∈
     pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)@i
38. ↑(snd(snd(snd(msgval(e'))))  < b)
⊢ w = v ∈ Cmd
BY
{ (skip{use pv11_p1_A5_C2}
   THEN Unfold `pv11_p1_decision` (-10)
   THEN ExRepD
   THEN (Assert ⌈loc(e) ↓∈ accpts⌉⋅
         THENA (RepUR ``pv11_p1_p2a'send`` (-2)
                THEN Try ((RWO "make-Msg-as-mk-msg" (-2) THENA Auto))
                THEN RewriteWithILF (-2)
                THEN Auto)
         )
   THEN (Assert ⌈0 < #(accpts)⌉⋅

         THENA (BLemma `bag-size-bag-member` THEN Auto THEN D 0 THEN InstConcl [⌈loc(e)⌉]⋅ THEN Auto)

)) }

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

31. make-msg-interface(d1;client;make-Msg([decision];<k, w>)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e')@i'

32. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())@i
33. ↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e'))))) b)@i
34. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
35. ∀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'))))

36. e1 : E
37. delay : ℤ
38. (e1 < e)
39. pv11_p1_p2a'send(Cmd;f) loc(e)
    <fst(msgval(e)), fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ∈
     pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)@i
40. ↑(snd(snd(snd(msgval(e'))))  < b)
41. loc(e) ↓∈ accpts
42. 0 < #(accpts)
⊢ w = v ∈ Cmd

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. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 19. e : E@i 20. \mforall{}e1:E ((e1 < e) {}\mRightarrow{} (\mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}k:\mBbbZ{}. \mforall{}v:Cmd. (pv11\_p1\_pvalue\{i:l\}(Cmd;ldrs$_{uid}$;accpts;es;e1;<b, k, v>f) {}\mRightarrow{} (\mforall{}e':E. \mforall{}w:Cmd. (pv11\_p1\_decision\{i:l\}(Cmd;f;accpts;ldrs;ldrs$_{uid}$;reps;es;\000Ce';k;w) {}\mRightarrow{} has-es-info-type(es;e';f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num()) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (snd(snd(snd(msgval(e')))\000C)) b)) {}\mRightarrow{} (w = v)))))) 21. b : pv11\_p1\_Ballot\_Num()@i 22. k : \mBbbZ{}@i 23. v : Cmd@i 24. pv11\_p1\_pvalue\{i:l\}(Cmd;ldrs$_{uid}$;accpts;es;e;<b, k, v>f)@i' 25. header(e) = ``pv11\_p1 p2a`` 26. <b, k, v> = (snd(msgval(e))) 27. e' : E@i 28. w : Cmd@i 29. pv11\_p1\_decision\{i:l\}(Cmd;f;accpts;ldrs;ldrs$_{uid}$;reps;es;e';k;w)@i' 30. has-es-info-type(es;e';f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())@i 31. \muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (snd(snd(snd(msgval(e'))))) b)@i 32. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) 33. \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')))) 34. e1 : E 35. delay : \mBbbZ{} 36. (e1 < e) 37. pv11\_p1\_p2a'send(Cmd;f) loc(e) <fst(msgval(e)), fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}$;reps;f)(e1)@i 38. \muparrow{}(snd(snd(snd(msgval(e')))) < b) \mvdash{} w = v By Latex: (skip\{use pv11\_p1\_A5\_C2\} THEN Unfold `pv11\_p1\_decision` (-10) THEN ExRepD THEN (Assert \mkleeneopen{}loc(e) \mdownarrow{}\mmember{} accpts\mkleeneclose{}\mcdot{} THENA (RepUR ``pv11\_p1\_p2a'send`` (-2) THEN Try ((RWO "make-Msg-as-mk-msg" (-2) THENA Auto)) THEN RewriteWithILF (-2) THEN Auto) ) THEN (Assert \mkleeneopen{}0 < \#(accpts)\mkleeneclose{}\mcdot{} THENA (BLemma `bag-size-bag-member` THEN Auto THEN D 0 THEN InstConcl [\mkleeneopen{}loc(e)\mkleeneclose{}]\mcdot{} THEN Auto) )) Home Index