Proof of Nuprl Lemma : pv11_p1_acc_state_from

Step * 1 1 1 5 of Lemma pv11_p1_acc_state_from_p2a

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. ldrs_uid : Id ─→ ℤ@i
14. Inj(Id;ℤ;ldrs_uid)@i
15. e : E@i
16. ∀e1:E
((e1 < e)
⇒

 (∀v1:pv11_p1_Ballot_Num(). ∀v2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List. ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀c:Cmd.

            (<v1, v2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e1)
            ⇒ (<b, s, c> ∈ v2)
            ⇒ (↓∃e':E
                  ∃l:Id
                   (e' ≤loc e1
                   ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e')

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

                            pv11_p1_on_p2a(Cmd;ldrs_uid);pv11_p1_p2a'base(Cmd;f))(pred(e'))@i

30. (<b, s, c> ∈ s2)
⇒ (↓∃e'@0:E
      ∃l:Id
       (e'@0 ≤loc pred(e')
       ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'@0)
       ∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0))) ∈ pv11_p1_Ballot_Num())
       ∧ (∀e'':E
            (e'@0 ≤loc e''
            ⇒ e'' ≤loc pred(e')
            ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'')))))))@i
31. ¬↑first(e')
32. ↑e' ∈_b pv11_p1_p2a'base(Cmd;f)
33. a : Id × pv11_p1_Ballot_Num() × ℤ × Cmd@i
34. a ∈ pv11_p1_p2a'base(Cmd;f)(e')@i
⊢ let v1,v2 = pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e') a <s1, s2>
  in (<b, s, c> ∈ v2)
     ⇒ (↓∃e'@0:E
           ∃l:Id
            (e'@0 ≤loc e'
            ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'@0)
            ∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0))) ∈ pv11_p1_Ballot_Num())
            ∧ (∀e'':E
                 (e'@0 ≤loc e''
                 ⇒ e'' ≤loc e'
                 ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'')))))))
BY
{ (All(Fold `pv11_p1_AcceptorState`)
   THEN DProdsAndUnions
   THEN RepUR ``pv11_p1_on_p2a`` 0
   THEN Auto
   THEN ExRepD
   THEN SplitOnHypITE (-1)
   THEN Auto
   THEN Try (All(\i.((RWO "pv11_p1_add_if_new_iff2" i THENA Auto) THEN DProdsAndUnions THEN Auto))⋅)
   THEN Try (ExRepD)) }

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

 (∀v1:pv11_p1_Ballot_Num(). ∀v2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List. ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀c:Cmd.

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

                   ∧ (∀e'':E
                        (e' ≤loc e''
                        ⇒ e'' ≤loc e1
                        ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e''))))))))))
17. v1 : pv11_p1_Ballot_Num()@i
18. v2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
19. b : pv11_p1_Ballot_Num()@i
20. s : ℤ@i
21. c : Cmd@i
22. <v1, v2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e)@i
23. (<b, s, c> ∈ v2)@i
24. s1 : pv11_p1_Ballot_Num()@i
25. s2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. e' : E@i
27. ¬↑e' ∈_b pv11_p1_p1a'base(Cmd;f)
28. e' ≤loc e @i
29. <s1, s2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(pred(e'))@i
30. ¬↑first(e')
31. ↑e' ∈_b pv11_p1_p2a'base(Cmd;f)
32. a1 : Id@i
33. a3 : pv11_p1_Ballot_Num()@i
34. a5 : ℤ@i
35. a6 : Cmd@i
36. <a1, a3, a5, a6> ∈ pv11_p1_p2a'base(Cmd;f)(e')@i
37. (<b, s, c> ∈ s2)
38. ↑(pv11_p1_leq_bnum(ldrs_uid) s1 a3)
39. e'@0 : E@i
40. l : Id@i
41. e'@0 ≤loc pred(e') @i
42. <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'@0)@i
43. b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0))) ∈ pv11_p1_Ballot_Num()@i
44. ∀e'':E
      (e'@0 ≤loc e''  ⇒ e'' ≤loc pred(e')  ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e''))))@i
⊢ ↓∃e'@0:E
    ∃l:Id
     (e'@0 ≤loc e'
     ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'@0)
     ∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0))) ∈ pv11_p1_Ballot_Num())
     ∧ (∀e'':E. (e'@0 ≤loc e''  ⇒ e'' ≤loc e'  ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e''))))))

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

 (∀v1:pv11_p1_Ballot_Num(). ∀v2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List. ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀c:Cmd.

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

                   ∧ (∀e'':E
                        (e' ≤loc e''
                        ⇒ e'' ≤loc e1
                        ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e''))))))))))
17. v1 : pv11_p1_Ballot_Num()@i
18. v2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
19. b : pv11_p1_Ballot_Num()@i
20. s : ℤ@i
21. c : Cmd@i
22. <v1, v2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e)@i
23. (<b, s, c> ∈ v2)@i
24. s1 : pv11_p1_Ballot_Num()@i
25. s2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. e' : E@i
27. ¬↑e' ∈_b pv11_p1_p1a'base(Cmd;f)
28. e' ≤loc e @i
29. <s1, s2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(pred(e'))@i
30. (<b, s, c> ∈ s2)
⇒ (↓∃e'@0:E
      ∃l:Id
       (e'@0 ≤loc pred(e')
       ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'@0)
       ∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0))) ∈ pv11_p1_Ballot_Num())
       ∧ (∀e'':E
            (e'@0 ≤loc e''
            ⇒ e'' ≤loc pred(e')
            ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'')))))))@i
31. ¬↑first(e')
32. ↑e' ∈_b pv11_p1_p2a'base(Cmd;f)
33. a1 : Id@i
34. a3 : pv11_p1_Ballot_Num()@i
35. a5 : ℤ@i
36. a6 : Cmd@i
37. <a1, a3, a5, a6> ∈ pv11_p1_p2a'base(Cmd;f)(e')@i

38. if (∃z∈s2.pv11_p1_same_pvalue(Cmd) <a3, a5, a6> z)_b then False else <b, s, c> = <a3, a5, a6> ∈ (pv11_p1_Ballot_Num(\000C) × ℤ × Cmd) fi

39. ↑(pv11_p1_leq_bnum(ldrs_uid) s1 a3)
⊢ ↓∃e'@0:E
    ∃l:Id
     (e'@0 ≤loc e'
     ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'@0)
     ∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0))) ∈ pv11_p1_Ballot_Num())
     ∧ (∀e'':E. (e'@0 ≤loc e''  ⇒ e'' ≤loc e'  ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e''))))))

3
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

 (∀v1:pv11_p1_Ballot_Num(). ∀v2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List. ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀c:Cmd.

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

                   ∧ (∀e'':E
                        (e' ≤loc e''
                        ⇒ e'' ≤loc e1
                        ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e''))))))))))
17. v1 : pv11_p1_Ballot_Num()@i
18. v2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
19. b : pv11_p1_Ballot_Num()@i
20. s : ℤ@i
21. c : Cmd@i
22. <v1, v2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e)@i
23. (<b, s, c> ∈ v2)@i
24. s1 : pv11_p1_Ballot_Num()@i
25. s2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. e' : E@i
27. ¬↑e' ∈_b pv11_p1_p1a'base(Cmd;f)
28. e' ≤loc e @i
29. <s1, s2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(pred(e'))@i
30. ¬↑first(e')
31. ↑e' ∈_b pv11_p1_p2a'base(Cmd;f)
32. a1 : Id@i
33. a3 : pv11_p1_Ballot_Num()@i
34. a5 : ℤ@i
35. a6 : Cmd@i
36. <a1, a3, a5, a6> ∈ pv11_p1_p2a'base(Cmd;f)(e')@i
37. (<b, s, c> ∈ s2)@i
38. ¬↑(pv11_p1_leq_bnum(ldrs_uid) s1 a3)
39. e'@0 : E@i
40. l : Id@i
41. e'@0 ≤loc pred(e') @i
42. <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'@0)@i
43. b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0))) ∈ pv11_p1_Ballot_Num()@i
44. ∀e'':E
      (e'@0 ≤loc e''  ⇒ e'' ≤loc pred(e')  ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e''))))@i
⊢ ↓∃e'@0:E
    ∃l:Id
     (e'@0 ≤loc e'
     ∧ <l, b, s, c> ∈ pv11_p1_p2a'base(Cmd;f)(e'@0)
     ∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'@0))) ∈ pv11_p1_Ballot_Num())
     ∧ (∀e'':E. (e'@0 ≤loc e''  ⇒ e'' ≤loc e'  ⇒ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e''))))))

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. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 14. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 15. e : E@i 16. \mforall{}e1:E ((e1 < e) {}\mRightarrow{} (\mforall{}v1:pv11\_p1\_Ballot\_Num(). \mforall{}v2:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}s:\mBbbZ{}. \mforall{}c:Cmd. (<v1, v2> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e1) {}\mRightarrow{} (<b, s, c> \mmember{} v2) {}\mRightarrow{} (\mdownarrow{}\mexists{}e':E \mexists{}l:Id (e' \mleq{}loc e1 \mwedge{} <l, b, s, c> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e') \mwedge{} (b = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e'\000C)))) \mwedge{} (\mforall{}e'':E (e' \mleq{}loc e'' {}\mRightarrow{} e'' \mleq{}loc e1 {}\mRightarrow{} (<b, s, c> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}\000C$;f;es;e'')))))))))) 17. v1 : pv11\_p1\_Ballot\_Num()@i 18. v2 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 19. b : pv11\_p1\_Ballot\_Num()@i 20. s : \mBbbZ{}@i 21. c : Cmd@i 22. <v1, v2> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e)@i 23. (<b, s, c> \mmember{} v2)@i 24. s1 : pv11\_p1\_Ballot\_Num()@i 25. s2 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 26. e' : E@i 27. \mneg{}\muparrow{}e' \mmember{}\msubb{} pv11\_p1\_p1a'base(Cmd;f) 28. e' \mleq{}loc e @i 29. <s1, s2> \mmember{} state-class2(\mlambda{}loc.pv11\_p1\_init\_acceptor(Cmd);pv11\_p1\_on\_p1a(Cmd;ldrs$_{ui\000Cd}$); pv11\_p1\_p1a'base(Cmd;f); pv11\_p1\_on\_p2a(Cmd;ldrs$_{uid}$);pv11\_p1\_p2a'base(Cm\000Cd;f))(pred(e'))@i 30. (<b, s, c> \mmember{} s2) {}\mRightarrow{} (\mdownarrow{}\mexists{}e'@0:E \mexists{}l:Id (e'@0 \mleq{}loc pred(e') \mwedge{} <l, b, s, c> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e'@0) \mwedge{} (b = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e'@0)))) \mwedge{} (\mforall{}e'':E (e'@0 \mleq{}loc e'' {}\mRightarrow{} e'' \mleq{}loc pred(e') {}\mRightarrow{} (<b, s, c> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e\000C'')))))))@i 31. \mneg{}\muparrow{}first(e') 32. \muparrow{}e' \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f) 33. a : Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd@i 34. a \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e')@i \mvdash{} let v1,v2 = pv11\_p1\_on\_p2a(Cmd;ldrs$_{uid}$) loc(e') a <s1, s2> in (<b, s, c> \mmember{} v2) {}\mRightarrow{} (\mdownarrow{}\mexists{}e'@0:E \mexists{}l:Id (e'@0 \mleq{}loc e' \mwedge{} <l, b, s, c> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e'@0) \mwedge{} (b = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e'@0)))) \mwedge{} (\mforall{}e'':E (e'@0 \mleq{}loc e'' {}\mRightarrow{} e'' \mleq{}loc e' {}\mRightarrow{} (<b, s, c> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f\000C;es;e''))))))) By Latex: (All(Fold `pv11\_p1\_AcceptorState`) THEN DProdsAndUnions THEN RepUR ``pv11\_p1\_on\_p2a`` 0 THEN Auto THEN ExRepD THEN SplitOnHypITE (-1) THEN Auto THEN Try (All(\mbackslash{}i.((RWO "pv11\_p1\_add\_if\_new\_iff2" i THENA Auto) THEN DProdsAndUnions THEN Auto))\mcdot{}) THEN Try (ExRepD)) Home Index