Proof of Nuprl Lemma : pv11_p1_scout_state_from

Step * 1 2 1 1 of Lemma pv11_p1_scout_state_from_p1bs

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. start : E@i
14. accpts : bag(Id)@i
15. ldrs : bag(Id)@i
16. ldrs_uid : Id ─→ ℤ@i
17. reps : bag(Id)@i
18. Inj(Id;ℤ;ldrs_uid)@i
19. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
20. e : E@i
21. ∀e1:E
      ((e1 < e)
      ⇒ start ≤loc e1
      ⇒ (∀bnum:pv11_p1_Ballot_Num(). ∀waitfor:bag(Id). ∀pvalues:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.
            ((<waitfor, pvalues>
            = pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;e1)
            ∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
            ⇒ (↓∃L:E List
                  ((waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id))
                  ∧ (∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
                       ((pv ∈ pvalues)
                       ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))))
                  ∧ (∀e':E
                       ((e' ∈ L)
                       ⇒ ((e' < e1)
                          ∧ loc(e') ↓∈ accpts
                          ∧ (bnum
                            = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))
                            ∈ pv11_p1_Ballot_Num())))))))))
22. start ≤loc e @i
23. bnum : pv11_p1_Ballot_Num()@i
24. waitfor : bag(Id)@i
25. pvalues : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. <waitfor, pvalues>
= <fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))) - fst(pv11_p1_p1b'base(Cmd;f)@e)

  , pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd) (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))

    (snd(snd(snd(pv11_p1_p1b'base(Cmd;f)@e))))
  >
∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
27. ↑e ∈_b Base(``pv11_p1 p1b``)
28. ¬↑first(e)
29. ¬↑first(e)
30. start ≤loc pred(e)
31. ↓∃L:E List
      (((fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
       = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))]
       ∈ bag(Id))
      ∧ (∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
           ((pv ∈ snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
           ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))))
      ∧ (∀e':E
           ((e' ∈ L)
           ⇒ ((e' < pred(e))
              ∧ loc(e') ↓∈ accpts

              ∧ (bnum = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())))))

32. pred(e) = pred(e) ∈ E
33. pred(e) = pred(e) ∈ E
34. pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
= pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
35. header(e) = ``pv11_p1 p1b`` ∈ Name
36. pv11_p1_p1b'base(Cmd;f)@e
= pv11_p1_p1b'base(Cmd;f)@e
∈ (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
37. bnum = (fst(snd(pv11_p1_p1b'base(Cmd;f)@e))) ∈ pv11_p1_Ballot_Num()
38. bnum = (fst(snd(snd(pv11_p1_p1b'base(Cmd;f)@e)))) ∈ pv11_p1_Ballot_Num()

39. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

⊢ ↓∃L:E List
    ((waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id))
    ∧ (∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
         ((pv ∈ pvalues) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))))
    ∧ (∀e':E
         ((e' ∈ L)
         ⇒ ((e' < e)
            ∧ loc(e') ↓∈ accpts

            ∧ (bnum = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())))))

BY
{ (SimpleSimpHyp Auto (-14)
   THEN UseMessageConstraint (-5)
   THEN Try (Complete ((Repeat ((RWO "pair-eta<" 0 THENA Auto)) THEN Auto)))
   THEN DProdsAndUnions
   THEN D 0
   THEN InstConcl [⌈L @ [e']⌉]⋅
   THEN Reduce 0
   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

26. waitfor = fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))) - fst(pv11_p1_p1b'base(Cmd;f)@e) ∈ bag(Id)

27. pvalues

= (pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd) (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))

(snd(snd(snd(pv11_p1_p1b'base(Cmd;f)@e)))))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
28. ↑e ∈_b Base(``pv11_p1 p1b``)
29. ¬↑first(e)
30. ¬↑first(e)
31. start ≤loc pred(e)
32. L : E List

33. (fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id)

34. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
      ((pv ∈ snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))
      ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L))))
35. ∀e':E
      ((e' ∈ L)
      ⇒ ((e' < pred(e))
         ∧ loc(e') ↓∈ accpts
         ∧ (bnum = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())))
36. pred(e) = pred(e) ∈ E
37. pred(e) = pred(e) ∈ E
38. pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
= pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
39. header(e) = ``pv11_p1 p1b`` ∈ Name
40. pv11_p1_p1b'base(Cmd;f)@e
= pv11_p1_p1b'base(Cmd;f)@e
∈ (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
41. bnum = (fst(snd(pv11_p1_p1b'base(Cmd;f)@e))) ∈ pv11_p1_Ballot_Num()
42. bnum = (fst(snd(snd(pv11_p1_p1b'base(Cmd;f)@e)))) ∈ pv11_p1_Ballot_Num()

43. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

44. ∀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'))))

45. e' : E
46. delay : ℤ
47. (e' < e)
48. loc(e') ↓∈ accpts
49. header(e') = ``pv11_p1 p1a`` ∈ Name
50. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num())
51. delay = 0 ∈ ℤ
52. loc(e) = (fst(msgval(e'))) ∈ Id
53. es-info-auth(es;e) = pv11_p1_init_active()
54. (fst(msgval(e))) = loc(e') ∈ Id
55. (fst(snd(msgval(e)))) = (snd(msgval(e'))) ∈ pv11_p1_Ballot_Num()
56. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⊢ waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L @ [e']))] ∈ bag(Id)

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

26. waitfor = fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))) - fst(pv11_p1_p1b'base(Cmd;f)@e) ∈ bag(Id)

27. pvalues

= (pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd) (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))

33. (fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id)

43. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

44. ∀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'))))

45. e' : E
46. delay : ℤ
47. (e' < e)
48. loc(e') ↓∈ accpts
49. header(e') = ``pv11_p1 p1a`` ∈ Name
50. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num())
51. delay = 0 ∈ ℤ
52. loc(e) = (fst(msgval(e'))) ∈ Id
53. es-info-auth(es;e) = pv11_p1_init_active()
54. (fst(msgval(e))) = loc(e') ∈ Id
55. (fst(snd(msgval(e)))) = (snd(msgval(e'))) ∈ pv11_p1_Ballot_Num()
56. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
57. waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L @ [e']))] ∈ bag(Id)
58. pv : pv11_p1_Ballot_Num() × ℤ × Cmd@i
59. (pv ∈ pvalues)@i
⊢ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L @ [e'])))

3
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

26. waitfor = fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))) - fst(pv11_p1_p1b'base(Cmd;f)@e) ∈ bag(Id)

27. pvalues

= (pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd) (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))

33. (fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id)

43. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

44. ∀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'))))

45. e' : E
46. delay : ℤ
47. (e' < e)
48. loc(e') ↓∈ accpts
49. header(e') = ``pv11_p1 p1a`` ∈ Name
50. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num())
51. delay = 0 ∈ ℤ
52. loc(e) = (fst(msgval(e'))) ∈ Id
53. es-info-auth(es;e) = pv11_p1_init_active()
54. (fst(msgval(e))) = loc(e') ∈ Id
55. (fst(snd(msgval(e)))) = (snd(msgval(e'))) ∈ pv11_p1_Ballot_Num()
56. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
57. waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L @ [e']))] ∈ bag(Id)
58. pv : pv11_p1_Ballot_Num() × ℤ × Cmd@i
59. (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L @ [e'])))@i
⊢ (pv ∈ pvalues)

4
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

26. waitfor = fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))) - fst(pv11_p1_p1b'base(Cmd;f)@e) ∈ bag(Id)

27. pvalues

= (pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd) (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))

33. (fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id)

43. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

44. ∀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'))))

45. e' : E
46. delay : ℤ
47. (e' < e)
48. loc(e') ↓∈ accpts
49. header(e') = ``pv11_p1 p1a`` ∈ Name
50. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num())
51. delay = 0 ∈ ℤ
52. loc(e) = (fst(msgval(e'))) ∈ Id
53. es-info-auth(es;e) = pv11_p1_init_active()
54. (fst(msgval(e))) = loc(e') ∈ Id
55. (fst(snd(msgval(e)))) = (snd(msgval(e'))) ∈ pv11_p1_Ballot_Num()
56. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
57. waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L @ [e']))] ∈ bag(Id)
58. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ pvalues) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L @ [e']))))
59. e'@0 : E@i
60. (e'@0 ∈ L @ [e'])@i
⊢ (e'@0 < e)

5
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

26. waitfor = fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))) - fst(pv11_p1_p1b'base(Cmd;f)@e) ∈ bag(Id)

27. pvalues

= (pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd) (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))

33. (fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id)

43. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

44. ∀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'))))

45. e' : E
46. delay : ℤ
47. (e' < e)
48. loc(e') ↓∈ accpts
49. header(e') = ``pv11_p1 p1a`` ∈ Name
50. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num())
51. delay = 0 ∈ ℤ
52. loc(e) = (fst(msgval(e'))) ∈ Id
53. es-info-auth(es;e) = pv11_p1_init_active()
54. (fst(msgval(e))) = loc(e') ∈ Id
55. (fst(snd(msgval(e)))) = (snd(msgval(e'))) ∈ pv11_p1_Ballot_Num()
56. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
57. waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L @ [e']))] ∈ bag(Id)
58. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ pvalues) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L @ [e']))))
59. e'@0 : E@i
60. (e'@0 ∈ L @ [e'])@i
61. (e'@0 < e)
⊢ loc(e'@0) ↓∈ accpts

6
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

26. waitfor = fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))) - fst(pv11_p1_p1b'base(Cmd;f)@e) ∈ bag(Id)

27. pvalues

= (pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd) (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))))

33. (fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id)

43. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

44. ∀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'))))

45. e' : E
46. delay : ℤ
47. (e' < e)
48. loc(e') ↓∈ accpts
49. header(e') = ``pv11_p1 p1a`` ∈ Name
50. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num())
51. delay = 0 ∈ ℤ
52. loc(e) = (fst(msgval(e'))) ∈ Id
53. es-info-auth(es;e) = pv11_p1_init_active()
54. (fst(msgval(e))) = loc(e') ∈ Id
55. (fst(snd(msgval(e)))) = (snd(msgval(e'))) ∈ pv11_p1_Ballot_Num()
56. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
57. waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L @ [e']))] ∈ bag(Id)
58. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ pvalues) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L @ [e']))))
59. e'@0 : E@i
60. (e'@0 ∈ L @ [e'])@i
61. (e'@0 < e)
62. loc(e'@0) ↓∈ accpts
⊢ bnum = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0))) ∈ pv11_p1_Ballot_Num()

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. start : E@i 14. accpts : bag(Id)@i 15. ldrs : bag(Id)@i 16. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 17. reps : bag(Id)@i 18. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 19. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 20. e : E@i 21. \mforall{}e1:E ((e1 < e) {}\mRightarrow{} start \mleq{}loc e1 {}\mRightarrow{} (\mforall{}bnum:pv11\_p1\_Ballot\_Num(). \mforall{}waitfor:bag(Id). \mforall{}pvalues:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List. ((<waitfor, pvalues> = pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;e1)) {}\mRightarrow{} (\mdownarrow{}\mexists{}L:E List ((waitfor = [i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))]) \mwedge{} (\mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd ((pv \mmember{} pvalues) \mLeftarrow{}{}\mRightarrow{} (pv \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{\000Cuid}$;f;es;e))); L))))) \mwedge{} (\mforall{}e':E ((e' \mmember{} L) {}\mRightarrow{} ((e' < e1) \mwedge{} loc(e') \mdownarrow{}\mmember{} accpts \mwedge{} (bnum = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}\mbackslash{}ff\000C24;f;es;e')))))))))))) 22. start \mleq{}loc e @i 23. bnum : pv11\_p1\_Ballot\_Num()@i 24. waitfor : bag(Id)@i 25. pvalues : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 26. <waitfor, pvalues> = <fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))) - fst(pv11\_p1\_p1b'base(Cmd;f)@e) , pv11\_p1\_append\_news(Cmd) pv11\_p1\_same\_pvalue(Cmd) (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) (snd(snd(snd(pv11\_p1\_p1b'base(Cmd;f)@e)))) > 27. \muparrow{}e \mmember{}\msubb{} Base(``pv11\_p1 p1b``) 28. \mneg{}\muparrow{}first(e) 29. \mneg{}\muparrow{}first(e) 30. start \mleq{}loc pred(e) 31. \mdownarrow{}\mexists{}L:E List (((fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) = [i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))]) \mwedge{} (\mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd ((pv \mmember{} snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))) \mLeftarrow{}{}\mRightarrow{} (pv \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}\mbackslash{}ff2\000C4;f;es;e)));L))))) \mwedge{} (\mforall{}e':E ((e' \mmember{} L) {}\mRightarrow{} ((e' < pred(e)) \mwedge{} loc(e') \mdownarrow{}\mmember{} accpts \mwedge{} (bnum = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e'))\000C)))))) 32. pred(e) = pred(e) 33. pred(e) = pred(e) 34. pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)) = pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)) 35. header(e) = ``pv11\_p1 p1b`` 36. pv11\_p1\_p1b'base(Cmd;f)@e = pv11\_p1\_p1b'base(Cmd;f)@e 37. bnum = (fst(snd(pv11\_p1\_p1b'base(Cmd;f)@e))) 38. bnum = (fst(snd(snd(pv11\_p1\_p1b'base(Cmd;f)@e)))) 39. 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)) \mvdash{} \mdownarrow{}\mexists{}L:E List ((waitfor = [i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))]) \mwedge{} (\mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd ((pv \mmember{} pvalues) \mLeftarrow{}{}\mRightarrow{} (pv \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;\000Cf;es;e)));L))))) \mwedge{} (\mforall{}e':E ((e' \mmember{} L) {}\mRightarrow{} ((e' < e) \mwedge{} loc(e') \mdownarrow{}\mmember{} accpts \mwedge{} (bnum = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e'))))\000C)))) By Latex: (SimpleSimpHyp Auto (-14) THEN UseMessageConstraint (-5) THEN Try (Complete ((Repeat ((RWO "pair-eta<" 0 THENA Auto)) THEN Auto))) THEN DProdsAndUnions THEN D 0 THEN InstConcl [\mkleeneopen{}L @ [e']\mkleeneclose{}]\mcdot{} THEN Reduce 0 THEN Auto) Home Index