Proof of Nuprl Lemma : pv11_p1_A4

Step * 1 2 4 of Lemma pv11_p1_A4_C1

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. Inj(Id;ℤ;ldrs_uid)@i
18. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
19. e2 : E@i
20. ∀e3:E
((e3 < e2)
⇒

 (∀e1:E. ∀bnum1,bnum2:pv11_p1_Ballot_Num(). ∀accepted1,accepted2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.

          ∀i1,i2,l:Id. ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀p1,p2:Cmd.
            (((<bnum1, accepted1> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e1)
            ∧ <bnum2, accepted2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e3)
            ∧ (<b, s, p1> ∈ accepted1)
            ∧ (<b, s, p2> ∈ accepted2))

            ∨ (pv11_p1_p2a'send(Cmd;f) i1 <l, b, s, p1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)

              ∧ pv11_p1_p2a'send(Cmd;f) i2 <l, b, s, p2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e3)))

⇒ (p1 = p2 ∈ Cmd))))
21. e1 : E@i
22. bnum1 : pv11_p1_Ballot_Num()@i
23. bnum2 : pv11_p1_Ballot_Num()@i
24. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
25. accepted2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. i1 : Id@i
27. i2 : Id@i
28. l : Id@i
29. b : pv11_p1_Ballot_Num()@i
30. s : ℤ@i
31. p1 : Cmd@i
32. p2 : Cmd@i
33. loc(e1) ↓∈ ldrs
34. i1 ↓∈ accpts
35. 0 = 0 ∈ ℤ
36. ff = pv11_p1_init_active()
37. l = loc(e1) ∈ Id
38. header(e1) = ``pv11_p1 adopted`` ∈ Name
39. has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
40. (fst(msgval(e1))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))) ∈ pv11_p1_Ballot_Num()
41. (<s, p1> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))
∧ (¬(∃p2:Cmd. (<s, p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e1)))))))
∨ (∃v2:pv11_p1_Ballot_Num()
(<v2, s, p1> ↓∈ snd(msgval(e1))

    ∧ (¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, s, z8> ∈ snd(msgval(e1))))))))

42. b = (fst(msgval(e1))) ∈ pv11_p1_Ballot_Num()
43. loc(e2) ↓∈ ldrs
44. i2 ↓∈ accpts
45. 0 = 0 ∈ ℤ
46. ff = pv11_p1_init_active()
47. l = loc(e2) ∈ Id
48. header(e2) = ``pv11_p1 adopted`` ∈ Name
49. has-es-info-type(es;e2;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
50. (fst(msgval(e2))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2))) ∈ pv11_p1_Ballot_Num()
51. (<s, p2> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2)))
∧ (¬(∃p2:Cmd. (<s, p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e2)))))))
∨ (∃v2:pv11_p1_Ballot_Num()
(<v2, s, p2> ↓∈ snd(msgval(e2))

    ∧ (¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, s, z8> ∈ snd(msgval(e2))))))))

52. b = (fst(msgval(e2))) ∈ pv11_p1_Ballot_Num()
53. loc(e1) = loc(e2) ∈ Id
⊢ p1 = p2 ∈ Cmd
BY
{ ((Assert (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))
          = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2)))
          ∈ pv11_p1_Ballot_Num() BY
          Auto)
   THEN (Assert <b, snd(msgval(e1))> ∈ pv11_p1_adopted'base(Cmd;f)(e1)
               ∧ <b, snd(msgval(e2))> ∈ pv11_p1_adopted'base(Cmd;f)(e2) BY
               (Auto
                THEN RepUR ``pv11_p1_adopted'base`` 0
                THEN BLemma `base-classrel-equal`
                THEN Auto
                THEN D 0
                THEN Auto
                THEN (CompleteGenConclAtAddr [3] THENA Auto)
                THEN D -2
                THEN Reduce 0
                THEN Auto))

   THEN (InstLemma `pv11_p1_unique_adopted_fun2` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈e1⌉;⌈e2⌉;⌈accpts⌉;⌈ldrs⌉;⌈ldrs_uid⌉;⌈reps⌉;⌈b⌉;

         ⌈snd(msgval(e1))⌉;⌈snd(msgval(e2))⌉]⋅
         THENA 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

 (∀e1:E. ∀bnum1,bnum2:pv11_p1_Ballot_Num(). ∀accepted1,accepted2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.

            ∨ (pv11_p1_p2a'send(Cmd;f) i1 <l, b, s, p1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)

              ∧ pv11_p1_p2a'send(Cmd;f) i2 <l, b, s, p2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e3)))

    ∧ (¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, s, z8> ∈ snd(msgval(e1))))))))

    ∧ (¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, s, z8> ∈ snd(msgval(e2))))))))

52. b = (fst(msgval(e2))) ∈ pv11_p1_Ballot_Num()
53. loc(e1) = loc(e2) ∈ Id
54. (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))
= (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2)))
∈ pv11_p1_Ballot_Num()

55. <b, snd(msgval(e1))> ∈ pv11_p1_adopted'base(Cmd;f)(e1) ∧ <b, snd(msgval(e2))> ∈ pv11_p1_adopted'base(Cmd;f)(e2)

56. (snd(msgval(e1))) = (snd(msgval(e2))) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
⊢ p1 = p2 ∈ 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. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 18. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 19. e2 : E@i 20. \mforall{}e3:E ((e3 < e2) {}\mRightarrow{} (\mforall{}e1:E. \mforall{}bnum1,bnum2:pv11\_p1\_Ballot\_Num(). \mforall{}accepted1,accepted2:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List. \mforall{}i1,i2,l:Id. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}s:\mBbbZ{}. \mforall{}p1,p2:Cmd. (((<bnum1, accepted1> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e1\000C) \mwedge{} <bnum2, accepted2> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e3) \mwedge{} (<b, s, p1> \mmember{} accepted1) \mwedge{} (<b, s, p2> \mmember{} accepted2)) \mvee{} (pv11\_p1\_p2a'send(Cmd;f) i1 <l, b, s, p1> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$\mbackslash{}ff5\000Cf{uid}$;reps;f)(e1) \mwedge{} pv11\_p1\_p2a'send(Cmd;f) i2 <l, b, s, p2> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$\mbackslash{}ff\000C5f{uid}$;reps;f)(e3))) {}\mRightarrow{} (p1 = p2)))) 21. e1 : E@i 22. bnum1 : pv11\_p1\_Ballot\_Num()@i 23. bnum2 : pv11\_p1\_Ballot\_Num()@i 24. accepted1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 25. accepted2 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 26. i1 : Id@i 27. i2 : Id@i 28. l : Id@i 29. b : pv11\_p1\_Ballot\_Num()@i 30. s : \mBbbZ{}@i 31. p1 : Cmd@i 32. p2 : Cmd@i 33. loc(e1) \mdownarrow{}\mmember{} ldrs 34. i1 \mdownarrow{}\mmember{} accpts 35. 0 = 0 36. ff = pv11\_p1\_init\_active() 37. l = loc(e1) 38. header(e1) = ``pv11\_p1 adopted`` 39. has-es-info-type(es;e1;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 40. (fst(msgval(e1))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1))) 41. (<s, p1> \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1))) \mwedge{} (\mneg{}(\mexists{}p2:Cmd. (<s, p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) (snd(msgval(e1))))))) \mvee{} (\mexists{}v2:pv11\_p1\_Ballot\_Num() (<v2, s, p1> \mdownarrow{}\mmember{} snd(msgval(e1)) \mwedge{} (\mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num(). \mexists{}z8:Cmd. ((\muparrow{}(v2 < z5)) \mwedge{} (<z5, s, z8> \mmember{} snd(msgval(e1)))))))) 42. b = (fst(msgval(e1))) 43. loc(e2) \mdownarrow{}\mmember{} ldrs 44. i2 \mdownarrow{}\mmember{} accpts 45. 0 = 0 46. ff = pv11\_p1\_init\_active() 47. l = loc(e2) 48. header(e2) = ``pv11\_p1 adopted`` 49. has-es-info-type(es;e2;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 50. (fst(msgval(e2))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e2))) 51. (<s, p2> \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e2))) \mwedge{} (\mneg{}(\mexists{}p2:Cmd. (<s, p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) (snd(msgval(e2))))))) \mvee{} (\mexists{}v2:pv11\_p1\_Ballot\_Num() (<v2, s, p2> \mdownarrow{}\mmember{} snd(msgval(e2)) \mwedge{} (\mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num(). \mexists{}z8:Cmd. ((\muparrow{}(v2 < z5)) \mwedge{} (<z5, s, z8> \mmember{} snd(msgval(e2)))))))) 52. b = (fst(msgval(e2))) 53. loc(e1) = loc(e2) \mvdash{} p1 = p2 By Latex: ((Assert (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e2))) BY Auto) THEN (Assert <b, snd(msgval(e1))> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e1) \mwedge{} <b, snd(msgval(e2))> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e2) BY (Auto THEN RepUR ``pv11\_p1\_adopted'base`` 0 THEN BLemma `base-classrel-equal` THEN Auto THEN D 0 THEN Auto THEN (CompleteGenConclAtAddr [3] THENA Auto) THEN D -2 THEN Reduce 0 THEN Auto)) THEN (InstLemma `pv11\_p1\_unique\_adopted\_fun2` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e1\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{};\mkleeneopen{}accpts\mkleeneclose{};\mkleeneopen{}ldrs\mkleeneclose{};\mkleeneopen{}ldrs$\mbackslash{}\000Cff5f{uid}$\mkleeneclose{}; \mkleeneopen{}reps\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}snd(msgval(e1))\mkleeneclose{};\mkleeneopen{}snd(msgval(e2))\mkleeneclose{}]\mcdot{} THENA Auto )) Home Index