Proof of Nuprl Lemma : pv11_p1_A5

Step * 1 2 2 1 1 2 of Lemma pv11_p1_A5_C2

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. b : pv11_p1_Ballot_Num()@i
18. s : ℤ@i
19. p : Cmd@i
20. p' : Cmd@i
21. as : bag(Id)@i
22. cs : bag(Id)@i
23. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
24. Inj(Id;ℤ;ldrs_uid)@i
25. accpts = (as + cs) ∈ bag(Id)@i
26. #(as) < pv11_p1_threshold(accpts)@i
27. ∀a:Id
(a ↓∈ cs ⇒

 (↓∃e':E. ((a = loc(e') ∈ Id) ∧ (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))))))@i

28. e : E@i
29. ∀e1:E
      ((e1 < e)
      ⇒ (∀b':pv11_p1_Ballot_Num()
            ((↑(b  < b'))
            ⇒ (∀l,i:Id.
                  (((<b', s, p'> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e1)))

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

                  ⇒ (p = p' ∈ Cmd))))))
30. b' : pv11_p1_Ballot_Num()@i
31. ↑(b  < b')@i
32. l : Id@i
33. i : Id@i
34. loc(e) ↓∈ ldrs
35. i ↓∈ accpts
36. 0 = 0 ∈ ℤ
37. ff = pv11_p1_init_active()
38. l = loc(e) ∈ Id
39. header(e) = ``pv11_p1 adopted`` ∈ Name
40. has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
41. (fst(msgval(e))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))) ∈ pv11_p1_Ballot_Num()
42. (<s, p'> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))
∧ (¬(∃p2:Cmd. (<s, p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e)))))))
∨ (∃v2:pv11_p1_Ballot_Num()
    (<v2, s, p'> ↓∈ snd(msgval(e))

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

43. b' = (fst(msgval(e))) ∈ pv11_p1_Ballot_Num()
44. <b', snd(msgval(e))> ∈ pv11_p1_adopted'base(Cmd;f)(e)
45. L : E List
46. #([i∈accpts|¬_bi ∈_b map(λe.loc(e);L))]) < pv11_p1_threshold(accpts)
47. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
      ((pv ∈ snd(msgval(e))) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L))))
48. ∀e':E
      ((e' ∈ L)
      ⇒ ((e' < e)
         ∧ loc(e') ↓∈ accpts
         ∧ (b' = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())))
49. i1 : Id
50. v : E
51. v ↓∈ L
52. i1 = loc(v) ∈ Id
53. i1 ↓∈ cs
54. e' : E
55. i1 = loc(e') ∈ Id
56. (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))
57. (v < e)
58. loc(v) ↓∈ accpts
59. b' = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v))) ∈ pv11_p1_Ballot_Num()
60. (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v)))
⊢ p = p' ∈ Cmd
BY
{ Assert ⌈(<b, s, p> ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))⌉⋅ }

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

 (↓∃e':E. ((a = loc(e') ∈ Id) ∧ (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))))))@i

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

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

43. b' = (fst(msgval(e))) ∈ pv11_p1_Ballot_Num()
44. <b', snd(msgval(e))> ∈ pv11_p1_adopted'base(Cmd;f)(e)
45. L : E List
46. #([i∈accpts|¬_bi ∈_b map(λe.loc(e);L))]) < pv11_p1_threshold(accpts)
47. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
      ((pv ∈ snd(msgval(e))) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L))))
48. ∀e':E
      ((e' ∈ L)
      ⇒ ((e' < e)
         ∧ loc(e') ↓∈ accpts
         ∧ (b' = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())))
49. i1 : Id
50. v : E
51. v ↓∈ L
52. i1 = loc(v) ∈ Id
53. i1 ↓∈ cs
54. e' : E
55. i1 = loc(e') ∈ Id
56. (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))
57. (v < e)
58. loc(v) ↓∈ accpts
59. b' = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v))) ∈ pv11_p1_Ballot_Num()
60. (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v)))
⊢ (<b, s, p> ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))

2
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

 (↓∃e':E. ((a = loc(e') ∈ Id) ∧ (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))))))@i

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

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

43. b' = (fst(msgval(e))) ∈ pv11_p1_Ballot_Num()
44. <b', snd(msgval(e))> ∈ pv11_p1_adopted'base(Cmd;f)(e)
45. L : E List
46. #([i∈accpts|¬_bi ∈_b map(λe.loc(e);L))]) < pv11_p1_threshold(accpts)
47. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
      ((pv ∈ snd(msgval(e))) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L))))
48. ∀e':E
      ((e' ∈ L)
      ⇒ ((e' < e)
         ∧ loc(e') ↓∈ accpts
         ∧ (b' = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())))
49. i1 : Id
50. v : E
51. v ↓∈ L
52. i1 = loc(v) ∈ Id
53. i1 ↓∈ cs
54. e' : E
55. i1 = loc(e') ∈ Id
56. (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))
57. (v < e)
58. loc(v) ↓∈ accpts
59. b' = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v))) ∈ pv11_p1_Ballot_Num()
60. (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v)))
61. (<b, s, p> ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))
⊢ p = p' ∈ 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. b : pv11\_p1\_Ballot\_Num()@i 18. s : \mBbbZ{}@i 19. p : Cmd@i 20. p' : Cmd@i 21. as : bag(Id)@i 22. cs : bag(Id)@i 23. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 24. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 25. accpts = (as + cs)@i 26. \#(as) < pv11\_p1\_threshold(accpts)@i 27. \mforall{}a:Id (a \mdownarrow{}\mmember{} cs {}\mRightarrow{} (\mdownarrow{}\mexists{}e':E ((a = loc(e')) \mwedge{} (<b, s, p> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid\mbackslash{}ff\000C7d$;f;es;e'))))))@i 28. e : E@i 29. \mforall{}e1:E ((e1 < e) {}\mRightarrow{} (\mforall{}b':pv11\_p1\_Ballot\_Num() ((\muparrow{}(b < b')) {}\mRightarrow{} (\mforall{}l,i:Id. (((<b', s, p'> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$\000C;f;es;e1))) \mvee{} pv11\_p1\_p2a'send(Cmd;f) i <l, b', s, p'> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs\mbackslash{}ff2\000C4_{uid}$;reps;f)(e1)) {}\mRightarrow{} (p = p')))))) 30. b' : pv11\_p1\_Ballot\_Num()@i 31. \muparrow{}(b < b')@i 32. l : Id@i 33. i : Id@i 34. loc(e) \mdownarrow{}\mmember{} ldrs 35. i \mdownarrow{}\mmember{} accpts 36. 0 = 0 37. ff = pv11\_p1\_init\_active() 38. l = loc(e) 39. header(e) = ``pv11\_p1 adopted`` 40. has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 41. (fst(msgval(e))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))) 42. (<s, p'> \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))) \mwedge{} (\mneg{}(\mexists{}p2:Cmd. (<s, p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) (snd(msgval(e))))))) \mvee{} (\mexists{}v2:pv11\_p1\_Ballot\_Num() (<v2, s, p'> \mdownarrow{}\mmember{} snd(msgval(e)) \mwedge{} (\mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num(). \mexists{}z8:Cmd. ((\muparrow{}(v2 < z5)) \mwedge{} (<z5, s, z8> \mmember{} snd(msgval(e)))))))) 43. b' = (fst(msgval(e))) 44. <b', snd(msgval(e))> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e) 45. L : E List 46. \#([i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))]) < pv11\_p1\_threshold(accpts) 47. \mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd ((pv \mmember{} snd(msgval(e))) \mLeftarrow{}{}\mRightarrow{} (pv \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;e\000Cs;e)));L)))) 48. \mforall{}e':E ((e' \mmember{} L) {}\mRightarrow{} ((e' < e) \mwedge{} loc(e') \mdownarrow{}\mmember{} accpts \mwedge{} (b' = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e')))))) 49. i1 : Id 50. v : E 51. v \mdownarrow{}\mmember{} L 52. i1 = loc(v) 53. i1 \mdownarrow{}\mmember{} cs 54. e' : E 55. i1 = loc(e') 56. (<b, s, p> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e'))) 57. (v < e) 58. loc(v) \mdownarrow{}\mmember{} accpts 59. b' = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;v))) 60. (<b, s, p> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;v))) \mvdash{} p = p' By Latex: Assert \mkleeneopen{}(<b, s, p> \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}\mbackslash{}ff2\000C4;f;es;e)));L)))\mkleeneclose{}\mcdot{} Home Index