Proof of Nuprl Lemma : pv11_p1_scout_from

Step * 4 of Lemma pv11_p1_scout_from_acc

.....antecedent.....
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

⊢ ∀a:Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List). ∀e':E.

  ∀s:bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List).
    (e' ≤loc e
    ⇒ a ∈ pv11_p1_p1b'base(Cmd;f)(e')
    ⇒ if first(e')
       then s ↓∈ (λloc.{pv11_p1_init_scout(Cmd;accpts)}) loc(e')

       else s ∈ State-loc-comb(λloc.{pv11_p1_init_scout(Cmd;accpts)};pv11_p1_on_p1b(Cmd) bnum;pv11_p1_p1b'base(Cmd;f))(

                pred(e'))
       fi
    ⇒ let waitfor,pvalues = s
       in ∀p:pv11_p1_Ballot_Num() × ℤ × Cmd
            ((p ∈ pvalues)
            ⇒ (∃e':E
                 ∃l:Id
                  ∃r:(pv11_p1_Ballot_Num() × ℤ × Cmd) List

                   (e' ≤loc e  ∧ <l, bnum, bnum, r> ∈ pv11_p1_p1b'base(Cmd;f)(e') ∧ (p ∈ r))))

    ⇒ let waitfor,pvalues = pv11_p1_on_p1b(Cmd) bnum loc(e') a s
       in ∀p:pv11_p1_Ballot_Num() × ℤ × Cmd
            ((p ∈ pvalues)
            ⇒ (∃e':E
                 ∃l:Id
                  ∃r:(pv11_p1_Ballot_Num() × ℤ × Cmd) List

                   (e' ≤loc e  ∧ <l, bnum, bnum, r> ∈ pv11_p1_p1b'base(Cmd;f)(e') ∧ (p ∈ r)))))

BY
{ (Reduce 0
   THEN Try ((UnivCD THENM (BagMemberD (-1) THEN HypSubst' (-1) 0)))
   THEN (UnivCD THENA Auto)
   THEN DVar `s1'
   THEN DVar `a'
   THEN DVar `a2'
   THEN DVar `a4'
   THEN All Reduce
   THEN ComputeWithCaseSplits [1] 0
   THEN RepeatFor 2 (AutoSplit)
   THEN All (RepUR ``let``)
   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

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. s : bag(Id) × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)@i
16. bnum : pv11_p1_Ballot_Num()@i
17. s ∈ State-loc-comb(λloc.{pv11_p1_init_scout(Cmd;accpts)};pv11_p1_on_p1b(Cmd) bnum;pv11_p1_p1b'base(Cmd;f))(e)
18. a1 : Id@i
19. a3 : pv11_p1_Ballot_Num()@i
20. a5 : pv11_p1_Ballot_Num()@i
21. a6 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
22. e' : E@i
23. s2 : bag(Id)@i
24. s3 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
25. e' ≤loc e @i
26. <a1, a3, a5, a6> ∈ pv11_p1_p1b'base(Cmd;f)(e')@i
27. if first(e')
then <s2, s3> ↓∈ {pv11_p1_init_scout(Cmd;accpts)}

else <s2, s3> ∈ State-loc-comb(λloc.{pv11_p1_init_scout(Cmd;accpts)};pv11_p1_on_p1b(Cmd) bnum;pv11_p1_p1b'base(Cmd;f))(

                pred(e'))
fi @i
28. ∀p:pv11_p1_Ballot_Num() × ℤ × Cmd
      ((p ∈ s3)
      ⇒ (∃e':E
           ∃l:Id
            ∃r:(pv11_p1_Ballot_Num() × ℤ × Cmd) List

             (e' ≤loc e  ∧ <l, bnum, bnum, r> ∈ pv11_p1_p1b'base(Cmd;f)(e') ∧ (p ∈ r))))@i

29. bnum = a3 ∈ pv11_p1_Ballot_Num()
30. bnum = a5 ∈ pv11_p1_Ballot_Num()
31. p : pv11_p1_Ballot_Num() × ℤ × Cmd@i
32. (p ∈ pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd) s3 a6)@i
⊢ ∃e':E
∃l:Id

    ∃r:(pv11_p1_Ballot_Num() × ℤ × Cmd) List. (e' ≤loc e  ∧ <l, bnum, bnum, r> ∈ pv11_p1_p1b'base(Cmd;f)(e') ∧ (p ∈ r))

Latex:

Latex:
.....antecedent..... 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. s : bag(Id) \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)@i 16. bnum : pv11\_p1\_Ballot\_Num()@i 17. s \mmember{} State-loc-comb(\mlambda{}loc.\{pv11\_p1\_init\_scout(Cmd;accpts)\};pv11\_p1\_on\_p1b(Cmd) bnum;pv11\_p1\_p1b'base(Cmd;f))(e) \mvdash{} \mforall{}a:Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List). \mforall{}e':E. \mforall{}s:bag(Id) \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List). (e' \mleq{}loc e {}\mRightarrow{} a \mmember{} pv11\_p1\_p1b'base(Cmd;f)(e') {}\mRightarrow{} if first(e') then s \mdownarrow{}\mmember{} (\mlambda{}loc.\{pv11\_p1\_init\_scout(Cmd;accpts)\}) loc(e') else s \mmember{} State-loc-comb(\mlambda{}loc.\{pv11\_p1\_init\_scout(Cmd;accpts)\};pv11\_p1\_on\_p1b(Cmd) bnum;pv11\_p1\_p1b'base(Cmd;f))( pred(e')) fi {}\mRightarrow{} let waitfor,pvalues = s in \mforall{}p:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd ((p \mmember{} pvalues) {}\mRightarrow{} (\mexists{}e':E \mexists{}l:Id \mexists{}r:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List (e' \mleq{}loc e \mwedge{} <l, bnum, bnum, r> \mmember{} pv11\_p1\_p1b'base(Cmd;f)(e') \mwedge{} (p \mmember{} r)))) {}\mRightarrow{} let waitfor,pvalues = pv11\_p1\_on\_p1b(Cmd) bnum loc(e') a s in \mforall{}p:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd ((p \mmember{} pvalues) {}\mRightarrow{} (\mexists{}e':E \mexists{}l:Id \mexists{}r:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List (e' \mleq{}loc e \mwedge{} <l, bnum, bnum, r> \mmember{} pv11\_p1\_p1b'base(Cmd;f)(e') \mwedge{} (p \mmember{} r))))) By Latex: (Reduce 0 THEN Try ((UnivCD THENM (BagMemberD (-1) THEN HypSubst' (-1) 0))) THEN (UnivCD THENA Auto) THEN DVar `s1' THEN DVar `a' THEN DVar `a2' THEN DVar `a4' THEN All Reduce THEN ComputeWithCaseSplits [1] 0 THEN RepeatFor 2 (AutoSplit) THEN All (RepUR ``let``) THEN Auto) Home Index