Proof of Nuprl Lemma : pv11_p1_scout_state_subset

Step * 1 2 1 of Lemma pv11_p1_scout_state_subset_pvals

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. e : E@i
14. ¬↑first(e)
15. bnum : pv11_p1_Ballot_Num()@i
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ─→ ℤ@i
19. reps : bag(Id)@i
20. Inj(Id;ℤ;ldrs_uid)@i
21. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
22. header(e) = ``pv11_p1 p1b`` ∈ Name@i
23. has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
24. (fst(snd(msgval(e)))) = bnum ∈ pv11_p1_Ballot_Num()@i
25. (fst(snd(snd(msgval(e))))) = bnum ∈ pv11_p1_Ballot_Num()@i
26. ↑e ∈_b pv11_p1_p1b'base(Cmd;f)

27. msgval(e) ∈ Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)

28. v1 : Id@i
29. v3 : pv11_p1_Ballot_Num()@i
30. v5 : pv11_p1_Ballot_Num()@i
31. v6 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
32. msgval(e)
= <v1, v3, v5, v6>
∈ (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
33. bnum = v3 ∈ pv11_p1_Ballot_Num()
34. bnum = v5 ∈ pv11_p1_Ballot_Num()
⊢ ↓l_subset(pv11_p1_Ballot_Num() × ℤ × Cmd;v6;pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd)

                                              (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es;pred(e))))

v6)
BY
{ (UseMessageConstraint (-13) THEN RepD) }

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

27. msgval(e) ∈ Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)

28. v1 : Id@i
29. v3 : pv11_p1_Ballot_Num()@i
30. v5 : pv11_p1_Ballot_Num()@i
31. v6 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
32. msgval(e)
= <v1, v3, v5, v6>
∈ (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
33. bnum = v3 ∈ pv11_p1_Ballot_Num()
34. bnum = v5 ∈ pv11_p1_Ballot_Num()
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. e' : E
37. delay : ℤ
38. (e' < e)
39. loc(e') ↓∈ accpts
40. header(e') = ``pv11_p1 p1a`` ∈ Name
41. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num())
42. delay = 0 ∈ ℤ
43. loc(e) = (fst(msgval(e'))) ∈ Id
44. es-info-auth(es;e) = pv11_p1_init_active()
45. (fst(msgval(e))) = loc(e') ∈ Id
46. (fst(snd(msgval(e)))) = (snd(msgval(e'))) ∈ pv11_p1_Ballot_Num()
47. <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))
⊢ ↓l_subset(pv11_p1_Ballot_Num() × ℤ × Cmd;v6;pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd)

                                              (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es;pred(e))))

v6)

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. e : E@i 14. \mneg{}\muparrow{}first(e) 15. bnum : pv11\_p1\_Ballot\_Num()@i 16. accpts : bag(Id)@i 17. ldrs : bag(Id)@i 18. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 19. reps : bag(Id)@i 20. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 21. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 22. header(e) = ``pv11\_p1 p1b``@i 23. 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))@i 24. (fst(snd(msgval(e)))) = bnum@i 25. (fst(snd(snd(msgval(e))))) = bnum@i 26. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p1b'base(Cmd;f) 27. msgval(e) \mmember{} Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List) 28. v1 : Id@i 29. v3 : pv11\_p1\_Ballot\_Num()@i 30. v5 : pv11\_p1\_Ballot\_Num()@i 31. v6 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 32. msgval(e) = <v1, v3, v5, v6>@i 33. bnum = v3 34. bnum = v5 \mvdash{} \mdownarrow{}l\_subset(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd;v6;pv11\_p1\_append\_news(Cmd) pv11\_p1\_same\_pvalue(Cmd) (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es;pred(e)))) v6) By Latex: (UseMessageConstraint (-13) THEN RepD) Home Index