Proof of Nuprl Lemma : pv11_p1_adopted_from_maj

Step * 1 1 of Lemma pv11_p1_adopted_from_maj_acceptors

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. 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. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)@i
23. e' : E
24. delay : ℤ
25. (e' < e)
26. loc(e') ↓∈ ldrs
27. ((no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e')) o pv11_p1_p1b'base(Cmd;f)) o
         pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e'))) prior to e')
∧ ((header(e') = ``pv11_p1 p1b`` ∈ Name)
  ∧ has-es-info-type(es;e';f;Id
    × pv11_p1_Ballot_Num()
    × pv11_p1_Ballot_Num()
    × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∧ ((pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num())
∧ (delay = 0 ∈ ℤ)
∧ (loc(e) = loc(e') ∈ Id)
∧ ((pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num())
∧ #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
∧ msg-authentic(info(e)) = pv11_p1_init_active()
∧ (bnum = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num())
∧ (pvals
  = (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
  ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∨ (∃e1:{e1:E| e1 ≤loc e' }
    ((((header(e1) = ``pv11_p1 preempted`` ∈ Name) ∧ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num()))
     ∧ (↑(pv11_p1_is_bnum() msgval(e1)))
     ∧ (↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))  < msgval(e1))))

    ∧ (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')
    ∧ ((header(e') = ``pv11_p1 p1b`` ∈ Name)
      ∧ has-es-info-type(es;e';f;Id
        × pv11_p1_Ballot_Num()
        × pv11_p1_Ballot_Num()
        × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
    ∧ ((pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num())
    ∧ (delay = 0 ∈ ℤ)
    ∧ (loc(e) = loc(e') ∈ Id)
    ∧ ((pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num())
    ∧ #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1)

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

    ∧ msg-authentic(info(e)) = pv11_p1_init_active()
    ∧ (bnum = (pv11_p1_upd_bnum() msgval(e1) loc(e1)) ∈ pv11_p1_Ballot_Num())
    ∧ (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))))
⊢ ↓∃L:E List
    (#([i∈accpts|¬_bi ∈_b map(λe.loc(e);L))]) < pv11_p1_threshold(accpts)
    ∧ (∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
         ((pv ∈ pvals) ⇐⇒ (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
{ ((Assert es-init(es;e') ≤loc e' BY EAuto 1) THEN PromoteHyp (-1) (-5)) }

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

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. 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. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)@i
23. e' : E
24. es-init(es;e') ≤loc e'
25. delay : ℤ
26. (e' < e)
27. loc(e') ↓∈ ldrs
28. ((no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e')) o pv11_p1_p1b'base(Cmd;f)) o
         pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e'))) prior to e')
∧ ((header(e') = ``pv11_p1 p1b`` ∈ Name)
  ∧ has-es-info-type(es;e';f;Id
    × pv11_p1_Ballot_Num()
    × pv11_p1_Ballot_Num()
    × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∧ ((pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num())
∧ (delay = 0 ∈ ℤ)
∧ (loc(e) = loc(e') ∈ Id)
∧ ((pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num())
∧ #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
∧ msg-authentic(info(e)) = pv11_p1_init_active()
∧ (bnum = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num())
∧ (pvals
  = (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
  ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
∨ (∃e1:{e1:E| e1 ≤loc e' }
    ((((header(e1) = ``pv11_p1 preempted`` ∈ Name) ∧ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num()))
     ∧ (↑(pv11_p1_is_bnum() msgval(e1)))
     ∧ (↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))  < msgval(e1))))

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

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

            ∧ (bnum = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ 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. 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. 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. <bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e)@i 23. e' : E 24. delay : \mBbbZ{} 25. (e' < e) 26. loc(e') \mdownarrow{}\mmember{} ldrs 27. ((no ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_init\_ballot\_num() loc(e')) o pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f) (pv11\_p1\_init\_ballot\_num() loc(e'))) prior to e') \mwedge{} ((header(e') = ``pv11\_p1 p1b``) \mwedge{} 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))) \mwedge{} ((pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(msgval(e'))))) \mwedge{} (delay = 0) \mwedge{} (loc(e) = loc(e')) \mwedge{} ((pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(snd(msgval(e')))))) \mwedge{} \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e'))) < pv11\_p1\_threshold(accpts) \mwedge{} msg-authentic(info(e)) = pv11\_p1\_init\_active() \mwedge{} (bnum = (pv11\_p1\_init\_ballot\_num() loc(e'))) \mwedge{} (pvals = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e'))))) \mvee{} (\mexists{}e1:\{e1:E| e1 \mleq{}loc e' \} ((((header(e1) = ``pv11\_p1 preempted``) \mwedge{} has-es-info-type(es;e1;f;pv11\_p1\_Ballot\_Num())) \mwedge{} (\muparrow{}(pv11\_p1\_is\_bnum() msgval(e1))) \mwedge{} (\muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)) < msgval(e1)))) \mwedge{} (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') \mwedge{} ((header(e') = ``pv11\_p1 p1b``) \mwedge{} 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))) \mwedge{} ((pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e'))))) \mwedge{} (delay = 0) \mwedge{} (loc(e) = loc(e')) \mwedge{} ((pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e')))))) \mwedge{} \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1) loc(e1);es.e1;e'))) < pv11\_p1\_threshold(accpts) \mwedge{} msg-authentic(info(e)) = pv11\_p1\_init\_active() \mwedge{} (bnum = (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1))) \mwedge{} (pvals = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1) loc(e1);es.e1;e')))))) \mvdash{} \mdownarrow{}\mexists{}L:E List (\#([i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))]) < pv11\_p1\_threshold(accpts) \mwedge{} (\mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd ((pv \mmember{} pvals) \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: ((Assert es-init(es;e') \mleq{}loc e' BY EAuto 1) THEN PromoteHyp (-1) (-5)) Home Index