Proof of Nuprl Lemma : pv11_p1_scout_from

Step * 4 1 of Lemma pv11_p1_scout_from_acc

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))

BY
{ (RepUR ``pv11_p1_append_news pv11_p1_add_if_new`` (-1)
   THEN (FLemma `member_list_accum_l_subset2` [-1]
         THENA (Auto
                THEN Unfold `l_subset` 0
                THEN Auto
                THEN (SplitOnHypITE (-1) THENA Auto)
                THEN GenListD 0
                THEN Try (Complete ((OrRight THEN Auto)))
                THEN GenListD (-2)
                THEN D (-2)
                THEN Try (Complete ((OrRight THEN Auto)))
                THEN GenListD (-2)
                THEN D (-2)
                THEN Try (Complete ((GenListD (-2) THEN Auto)))
                THEN OrLeft
                THEN Auto)
         )
   THEN D (-1)) }

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

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 ∈ accumulate (with value a and list item x):
          if (∃zzc∈a.pv11_p1_same_pvalue(Cmd) x zzc)_b then a else a @ [x] fi
         over list:
           a6
         with starting value:
          s3))@i
33. (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))

2
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

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

    ∃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:
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) 18. a1 : Id@i 19. a3 : pv11\_p1\_Ballot\_Num()@i 20. a5 : pv11\_p1\_Ballot\_Num()@i 21. a6 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 22. e' : E@i 23. s2 : bag(Id)@i 24. s3 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 25. e' \mleq{}loc e @i 26. <a1, a3, a5, a6> \mmember{} pv11\_p1\_p1b'base(Cmd;f)(e')@i 27. if first(e') then <s2, s3> \mdownarrow{}\mmember{} \{pv11\_p1\_init\_scout(Cmd;accpts)\} else <s2, s3> \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 @i 28. \mforall{}p:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd ((p \mmember{} s3) {}\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))))@i 29. bnum = a3 30. bnum = a5 31. p : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd@i 32. (p \mmember{} pv11\_p1\_append\_news(Cmd) pv11\_p1\_same\_pvalue(Cmd) s3 a6)@i \mvdash{} \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: (RepUR ``pv11\_p1\_append\_news pv11\_p1\_add\_if\_new`` (-1) THEN (FLemma `member\_list\_accum\_l\_subset2` [-1] THENA (Auto THEN Unfold `l\_subset` 0 THEN Auto THEN (SplitOnHypITE (-1) THENA Auto) THEN GenListD 0 THEN Try (Complete ((OrRight THEN Auto))) THEN GenListD (-2) THEN D (-2) THEN Try (Complete ((OrRight THEN Auto))) THEN GenListD (-2) THEN D (-2) THEN Try (Complete ((GenListD (-2) THEN Auto))) THEN OrLeft THEN Auto) ) THEN D (-1)) Home Index