Proof of Nuprl Lemma : pv11_p1_scout_state_from

Step * 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

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

BY
{ (MoveToConcl (-1)
   THEN RepeatFor 3 (MoveToConcl (-4))
   THEN MoveToConcl (-8)
   THEN CausalInd'
   THEN Auto
   THEN (RWO "pv11_p1_scout_state_fun_eq" (-1) THENA Auto)
   THEN (SplitOnHypITE (-1) THENA Auto)
   THEN (SplitOnHypITE (-2) THENA Auto)) }

1
.....truecase.....
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>
= (pv11_p1_on_p1b(Cmd) bnum loc(e) pv11_p1_p1b'base(Cmd;f)@e pv11_p1_init_scout(Cmd;accpts))
∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
27. ↑e ∈_b pv11_p1_p1b'base(Cmd;f)
28. ↑first(e)
⊢ ↓∃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())))))

2
.....falsecase.....
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

= (pv11_p1_on_p1b(Cmd) bnum loc(e) pv11_p1_p1b'base(Cmd;f)@e pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e)))

∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
27. ↑e ∈_b pv11_p1_p1b'base(Cmd;f)
28. ¬↑first(e)
⊢ ↓∃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())))))

3
.....truecase.....
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, pvalues> = pv11_p1_init_scout(Cmd;accpts) ∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

27. ¬↑e ∈_b pv11_p1_p1b'base(Cmd;f)
28. ↑first(e)
⊢ ↓∃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())))))

4
.....falsecase.....
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>
= pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es.start;pred(e))
∈ (bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
27. ¬↑e ∈_b pv11_p1_p1b'base(Cmd;f)
28. ¬↑first(e)
⊢ ↓∃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())))))

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. e : E@i 15. accpts : bag(Id)@i 16. ldrs : bag(Id)@i 17. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 18. reps : bag(Id)@i 19. bnum : pv11\_p1\_Ballot\_Num()@i 20. waitfor : bag(Id)@i 21. pvalues : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 22. start \mleq{}loc e @i 23. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 24. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 25. <waitfor, pvalues> = pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es.start;e)@i \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: (MoveToConcl (-1) THEN RepeatFor 3 (MoveToConcl (-4)) THEN MoveToConcl (-8) THEN CausalInd' THEN Auto THEN (RWO "pv11\_p1\_scout\_state\_fun\_eq" (-1) THENA Auto) THEN (SplitOnHypITE (-1) THENA Auto) THEN (SplitOnHypITE (-2) THENA Auto)) Home Index