Proof of Nuprl Lemma : pv11_p1_adopted_from_init_or

Step * 2 of Lemma pv11_p1_adopted_from_init_or_preempted

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. accpts : bag(Id)@i
15. ldrs : bag(Id)@i
16. ldrs_uid : Id ─→ ℤ@i
17. reps : bag(Id)@i
18. bnum : pv11_p1_Ballot_Num()@i
19. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
20. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
21. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)@i
22. e' : E
23. delay : ℤ
24. (e' < e)
25. loc(e') ↓∈ ldrs
26. e1 : {e1:E| e1 ≤loc e' }
27. header(e1) = ``pv11_p1 preempted`` ∈ Name
28. has-es-info-type(es;e1;f;pv11_p1_Ballot_Num())
29. ↑(pv11_p1_is_bnum() msgval(e1))
30. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)) < msgval(e1))

31. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1)) o pv11_p1_p1b'base(Cmd;f)) o

pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1))) between e1 and e')
32. header(e') = ``pv11_p1 p1b`` ∈ Name

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

34. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
35. delay = 0 ∈ ℤ
36. loc(e) = loc(e') ∈ Id
37. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
38. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1)

                                             loc(e1);es.e1;e'))) < pv11_p1_threshold(accpts)

39. msg-authentic(info(e)) = pv11_p1_init_active()
40. bnum = (pv11_p1_upd_bnum() msgval(e1) loc(e1)) ∈ pv11_p1_Ballot_Num()
41. pvals
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1) loc(e1);es.e1;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
⊢ ↓(bnum = (pv11_p1_init_ballot_num() loc(e)) ∈ pv11_p1_Ballot_Num())
   ∨ (∃e':E
       ∃bnum1,bnum2:pv11_p1_Ballot_Num()
        ∃b:𝔹
         ∃proposals:(ℤ × Cmd) List
          ((e' <loc e)
          ∧ bnum1 ∈ pv11_p1_preempted'base(Cmd;f)(e')
          ∧ <bnum2, b, proposals> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e')
          ∧ (↑(bnum2  < bnum1))
          ∧ (bnum = (pv11_p1_upd_bnum() bnum1 loc(e)) ∈ pv11_p1_Ballot_Num())))
BY
{ (DVarSets
   THEN D 0
   THEN (OrRight THENA Auto)
   THEN InstConcl [⌈e1⌉;⌈msgval(e1)⌉;⌈fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))⌉;

   ⌈fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))⌉;⌈snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))⌉]⋅

THEN Auto) }

1
1. Cmd : Type@i'
2. valueall-type(Cmd)@i'
3. f : pv11_p1_headers_type{i:l}(Cmd)@i'
4. (f [decision]) = (ℤ × Cmd) ∈ Type
5. (f [propose]) = (ℤ × Cmd) ∈ Type

6. (f ``pv11_p1 adopted``) = (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)) ∈ Type

7. (f ``pv11_p1 preempted``) = pv11_p1_Ballot_Num() ∈ Type
8. (f ``pv11_p1 p2b``) = (Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num()) ∈ Type
9. (f ``pv11_p1 p2a``) = (Id × pv11_p1_Ballot_Num() × ℤ × Cmd) ∈ Type
10. (f ``pv11_p1 p1b``)
= (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
∈ Type
11. (f ``pv11_p1 p1a``) = (Id × pv11_p1_Ballot_Num()) ∈ Type
12. f ∈ Name ─→ Type
13. es : EO+(Message(f))@i'
14. e : E@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs_uid : Id ─→ ℤ@i
18. reps : bag(Id)@i
19. bnum : pv11_p1_Ballot_Num()@i
20. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
21. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
22. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)@i
23. e' : E
24. delay : ℤ
25. (e' < e)
26. loc(e') ↓∈ ldrs
27. e1 : E
28. e1 ≤loc e'
29. header(e1) = ``pv11_p1 preempted`` ∈ Name
30. has-es-info-type(es;e1;f;pv11_p1_Ballot_Num())
31. ↑(pv11_p1_is_bnum() msgval(e1))
32. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)) < msgval(e1))

33. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1)) o pv11_p1_p1b'base(Cmd;f)) o

pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1))) between e1 and e')
34. header(e') = ``pv11_p1 p1b`` ∈ Name

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

36. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
37. delay = 0 ∈ ℤ
38. loc(e) = loc(e') ∈ Id
39. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
40. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1)

                                             loc(e1);es.e1;e'))) < pv11_p1_threshold(accpts)

41. msg-authentic(info(e)) = pv11_p1_init_active()
42. bnum = (pv11_p1_upd_bnum() msgval(e1) loc(e1)) ∈ pv11_p1_Ballot_Num()
43. pvals
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1) loc(e1);es.e1;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
44. (e1 <loc e)
45. msgval(e1) ∈ pv11_p1_Ballot_Num()
⊢ msgval(e1) ∈ pv11_p1_preempted'base(Cmd;f)(e1)

2
1. Cmd : Type@i'
2. valueall-type(Cmd)@i'
3. f : pv11_p1_headers_type{i:l}(Cmd)@i'
4. (f [decision]) = (ℤ × Cmd) ∈ Type
5. (f [propose]) = (ℤ × Cmd) ∈ Type

6. (f ``pv11_p1 adopted``) = (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)) ∈ Type

33. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1)) o pv11_p1_p1b'base(Cmd;f)) o

pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1))) between e1 and e')
34. header(e') = ``pv11_p1 p1b`` ∈ Name

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

                                             loc(e1);es.e1;e'))) < pv11_p1_threshold(accpts)

41. msg-authentic(info(e)) = pv11_p1_init_active()
42. bnum = (pv11_p1_upd_bnum() msgval(e1) loc(e1)) ∈ pv11_p1_Ballot_Num()
43. pvals
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1) loc(e1);es.e1;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
44. (e1 <loc e)
45. msgval(e1) ∈ pv11_p1_preempted'base(Cmd;f)(e1)
⊢ <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))
, fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))
, snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)

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. accpts : bag(Id)@i 15. ldrs : bag(Id)@i 16. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 17. reps : bag(Id)@i 18. bnum : pv11\_p1\_Ballot\_Num()@i 19. pvals : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 20. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 21. <bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e)@i 22. e' : E 23. delay : \mBbbZ{} 24. (e' < e) 25. loc(e') \mdownarrow{}\mmember{} ldrs 26. e1 : \{e1:E| e1 \mleq{}loc e' \} 27. header(e1) = ``pv11\_p1 preempted`` 28. has-es-info-type(es;e1;f;pv11\_p1\_Ballot\_Num()) 29. \muparrow{}(pv11\_p1\_is\_bnum() msgval(e1)) 30. \muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)) < msgval(e1)) 31. (no ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) o pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1))) between e1 and e') 32. header(e') = ``pv11\_p1 p1b`` 33. 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)) 34. (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e')))) 35. delay = 0 36. loc(e) = loc(e') 37. (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e'))))) 38. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1) loc(e1);es.e1;e'))) < pv11\_p1\_threshold(accpts) 39. msg-authentic(info(e)) = pv11\_p1\_init\_active() 40. bnum = (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) 41. pvals = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1) loc(e1);es.e1;e'))) \mvdash{} \mdownarrow{}(bnum = (pv11\_p1\_init\_ballot\_num() loc(e))) \mvee{} (\mexists{}e':E \mexists{}bnum1,bnum2:pv11\_p1\_Ballot\_Num() \mexists{}b:\mBbbB{} \mexists{}proposals:(\mBbbZ{} \mtimes{} Cmd) List ((e' <loc e) \mwedge{} bnum1 \mmember{} pv11\_p1\_preempted'base(Cmd;f)(e') \mwedge{} <bnum2, b, proposals> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e') \mwedge{} (\muparrow{}(bnum2 < bnum1)) \mwedge{} (bnum = (pv11\_p1\_upd\_bnum() bnum1 loc(e))))) By Latex: (DVarSets THEN D 0 THEN (OrRight THENA Auto) THEN InstConcl [\mkleeneopen{}e1\mkleeneclose{};\mkleeneopen{}msgval(e1)\mkleeneclose{};\mkleeneopen{}fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;\000Cf;es;e1))\mkleeneclose{}; \mkleeneopen{}fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)))\mkleeneclose{}; \mkleeneopen{}snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)))\mkleeneclose{}]\mcdot{} THEN Auto) Home Index