Proof of Nuprl Lemma : pv11_p1

Step * 1 1 2 1 of Lemma pv11_p1_A2

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. ¬↑first(e)
15. ¬↑e ∈_b pv11_p1_p1a'base(Cmd;f)
16. ldrs_uid : Id ─→ ℤ@i
17. bnum1 : pv11_p1_Ballot_Num()@i
18. bnum2 : pv11_p1_Ballot_Num()@i
19. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
20. b : pv11_p1_Ballot_Num()@i
21. pv : ℤ × Cmd@i
22. loc : Id@i
23. Inj(Id;ℤ;ldrs_uid)@i
24. ¬False@i
25. <bnum1, accepted1>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
26. ¬(<b, pv> ∈ accepted1)@i
27. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 bnum2)
28. l : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
29. (<b, pv> ∈ l)
30. ↑e ∈_b pv11_p1_p2a'base(Cmd;f)
31. pv11_p1_p2a'base(Cmd;f)@e ~ msgval(e)
⊢ (<bnum2, accepted1 @ l>
= let loc,zg = msgval(e)
  in let b,sp = zg
     in let ballot_num,accepted = pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
        in let ballot_num' = pv11_p1_max_bnum(ldrs_uid) b ballot_num in
            let accepted' = if pv11_p1_leq_bnum(ldrs_uid) ballot_num b

                            then pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <b, sp> accepted

                            else accepted
                            fi  in
            <ballot_num', accepted'>
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))
⇒ (b = bnum2 ∈ pv11_p1_Ballot_Num())
BY
{ (Unfold `pv11_p1_p2a\'base` (-2)
   THEN (FLemma `member-base-class` [-2] THENA Auto)
   THEN (Assert ⌈has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)⌉⋅
         THENA ((RWO "has-es-info-type-is-msg-has-type" 0 THENA Auto)
                THEN RepUR ``msg-has-type msg-type`` 0
                THEN Fold `es-header` 0
                THEN Auto)
         )
   THEN (GenConclAtAddrType ⌈Id × pv11_p1_Ballot_Num() × ℤ × Cmd⌉ [1;3;1]⋅ THENA Auto)
   THEN DProdsAndUnions
   THEN RepUR ``let`` 0
   THEN UsePairEta [1;3;1] 0
   THEN (D 0 THENA Auto)
   THEN SimpEqPairs
   THEN (RevHypSubst (-18) (-2) THENA Auto)
   THEN (RevHypSubst (-18) (-1) THENA Auto)
   THEN (RevHypSubst (-17) (-1) THENA Auto)
   THEN (SplitOnHypITE (-1) 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

27. accepted1 = (snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)

28. ¬(<b, p1, p2> ∈ accepted1)@i
29. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 bnum2)
30. l : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
31. (<b, p1, p2> ∈ l)
32. ↑e ∈_b Base(``pv11_p1 p2a``)
33. pv11_p1_p2a'base(Cmd;f)@e ~ msgval(e)
34. header(e) = ``pv11_p1 p2a`` ∈ Name
35. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
36. v1 : Id@i
37. v3 : pv11_p1_Ballot_Num()@i
38. v5 : ℤ@i
39. v6 : Cmd@i
40. (fst(msgval(e))) = v1 ∈ Id
41. (snd(msgval(e))) = <v3, v5, v6> ∈ (pv11_p1_Ballot_Num() × ℤ × Cmd)
42. bnum2 = (pv11_p1_max_bnum(ldrs_uid) v3 bnum1) ∈ pv11_p1_Ballot_Num()

43. (accepted1 @ l) = (pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd) <v3, v5, v6> accepted1) ∈ ((pv11_p1_Ballot_Num() × \000Cℤ × Cmd) List)

44. ↑(pv11_p1_leq_bnum(ldrs_uid) bnum1 v3)
⊢ b = bnum2 ∈ 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

27. accepted1 = (snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)

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. \mneg{}\muparrow{}first(e) 15. \mneg{}\muparrow{}e \mmember{}\msubb{} pv11\_p1\_p1a'base(Cmd;f) 16. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 17. bnum1 : pv11\_p1\_Ballot\_Num()@i 18. bnum2 : pv11\_p1\_Ballot\_Num()@i 19. accepted1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 20. b : pv11\_p1\_Ballot\_Num()@i 21. pv : \mBbbZ{} \mtimes{} Cmd@i 22. loc : Id@i 23. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 24. \mneg{}False@i 25. <bnum1, accepted1> = pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e)) 26. \mneg{}(<b, pv> \mmember{} accepted1)@i 27. \muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) bnum1 bnum2) 28. l : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List 29. (<b, pv> \mmember{} l) 30. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f) 31. pv11\_p1\_p2a'base(Cmd;f)@e \msim{} msgval(e) \mvdash{} (<bnum2, accepted1 @ l> = let loc,zg = msgval(e) in let b,sp = zg in let ballot$_{num}$,accepted = pv11\_p1\_AcceptorStateFun(Cmd;ldrs$\mbackslash{}ff5\000Cf{uid}$;f;es;pred(e)) in let ballot$_{num'}$ = pv11\_p1\_max\_bnum(ldrs$_{uid}\mbackslash{}ff\000C24) b ballot$_{num}$ in let accepted' = if pv11\_p1\_leq\_bnum(ldrs$_{uid}$) ballot$_\mbackslash{}f\000Cf7bnum}$ b then pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) <b, sp> accepted else accepted fi in <ballot$_{num'}$, accepted'>) {}\mRightarrow{} (b = bnum2) By Latex: (Unfold `pv11\_p1\_p2a\mbackslash{}'base` (-2) THEN (FLemma `member-base-class` [-2] THENA Auto) THEN (Assert \mkleeneopen{}has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)\mkleeneclose{}\mcdot{} THENA ((RWO "has-es-info-type-is-msg-has-type" 0 THENA Auto) THEN RepUR ``msg-has-type msg-type`` 0 THEN Fold `es-header` 0 THEN Auto) ) THEN (GenConclAtAddrType \mkleeneopen{}Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd\mkleeneclose{} [1;3;1]\mcdot{} THENA Auto) THEN DProdsAndUnions THEN RepUR ``let`` 0 THEN UsePairEta [1;3;1] 0 THEN (D 0 THENA Auto) THEN SimpEqPairs THEN (RevHypSubst (-18) (-2) THENA Auto) THEN (RevHypSubst (-18) (-1) THENA Auto) THEN (RevHypSubst (-17) (-1) THENA Auto) THEN (SplitOnHypITE (-1) THENA Auto)) Home Index