Proof of Nuprl Lemma : pv11_p1_A5

Step * 1 2 1 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) = [propose] ∈ Name
40. has-es-info-type(es;e;f;ℤ × Cmd)
41. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))))
42. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))))
43. b' = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))) ∈ pv11_p1_Ballot_Num()
44. <s, p'> = msgval(e) ∈ (ℤ × Cmd)
⊢ p = p' ∈ Cmd
BY
{ ((FLemma `pv11_p1_ldr_active_fun` [-4] THENA Auto)
   THEN SquashExRepD
   THEN (FLemma `pv11_p1_adopted_from_maj_acceptors` [23;-2]⋅ THENA Auto)
   THEN SquashExRepD
   THEN (InstLemma `pv11_p1_overlapping_accs` [⌈accpts⌉;⌈as⌉;⌈map(λe.loc(e);L)⌉;⌈cs⌉]⋅
         THENA (Auto THEN RepUR ``bag-deq-member`` 0 THEN Auto)
         )
   THEN SquashExRepD
   THEN TryBagMemberD
   THEN SimpleInstHyps
   THEN SquashExRepD
   THEN (InstHyp [⌈v⌉] (-9)⋅ THENA (Auto THEN BLemma `bag-member-list` THEN Auto))
   THEN RepD
   THEN (Assert b' = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v))) ∈ pv11_p1_Ballot_Num() BY
               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

 (↓∃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))

                  ⇒ (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) = [propose] ∈ Name
40. has-es-info-type(es;e;f;ℤ × Cmd)
41. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))))
42. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))))
43. b' = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))) ∈ pv11_p1_Ballot_Num()
44. <s, p'> = msgval(e) ∈ (ℤ × Cmd)
45. e' : E
46. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
47. (e' <loc e)
48. <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)), pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e')
49. (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))
= (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))
∈ pv11_p1_Ballot_Num()
50. L : E List
51. #([i∈accpts|¬_bi ∈_b map(λe.loc(e);L))]) < pv11_p1_threshold(accpts)
52. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
      ((pv ∈ pvals) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L))))
53. ∀e1:E
      ((e1 ∈ L)
      ⇒ ((e1 < e')
         ∧ loc(e1) ↓∈ accpts
         ∧ ((fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))
           = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e1)))
           ∈ pv11_p1_Ballot_Num())))
54. i1 : Id
55. v : E
56. v ↓∈ L
57. i1 = loc(v) ∈ Id
58. i1 ↓∈ cs
59. e1 : E
60. i1 = loc(e1) ∈ Id
61. (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e1)))
62. (v < e')
63. loc(v) ↓∈ accpts
64. (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))
= (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v)))
∈ pv11_p1_Ballot_Num()
65. b' = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v))) ∈ pv11_p1_Ballot_Num()
⊢ 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}$;reps;f)(e1)) {}\mRightarrow{} (p = p')))))) 30. b' : pv11\_p1\_Ballot\_Num()@i 31. \muparrow{}(b < b')@i 32. l : Id@i 33. i : Id@i 34. loc(e) \mdownarrow{}\mmember{} ldrs 35. i \mdownarrow{}\mmember{} accpts 36. 0 = 0 37. ff = pv11\_p1\_init\_active() 38. l = loc(e) 39. header(e) = [propose] 40. has-es-info-type(es;e;f;\mBbbZ{} \mtimes{} Cmd) 41. \muparrow{}(fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))) 42. \mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) (fst(msgval(e))) (snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))))) 43. b' = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))) 44. <s, p'> = msgval(e) \mvdash{} p = p' By Latex: ((FLemma `pv11\_p1\_ldr\_active\_fun` [-4] THENA Auto) THEN SquashExRepD THEN (FLemma `pv11\_p1\_adopted\_from\_maj\_acceptors` [23;-2]\mcdot{} THENA Auto) THEN SquashExRepD THEN (InstLemma `pv11\_p1\_overlapping\_accs` [\mkleeneopen{}accpts\mkleeneclose{};\mkleeneopen{}as\mkleeneclose{};\mkleeneopen{}map(\mlambda{}e.loc(e);L)\mkleeneclose{};\mkleeneopen{}cs\mkleeneclose{}]\mcdot{} THENA (Auto THEN RepUR ``bag-deq-member`` 0 THEN Auto) ) THEN SquashExRepD THEN TryBagMemberD THEN SimpleInstHyps THEN SquashExRepD THEN (InstHyp [\mkleeneopen{}v\mkleeneclose{}] (-9)\mcdot{} THENA (Auto THEN BLemma `bag-member-list` THEN Auto)) THEN RepD THEN (Assert b' = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;v))) BY Auto)) Home Index