Proof of Nuprl Lemma : pv11_p1_commander_state_from

Step * 1 1 of Lemma pv11_p1_commander_state_from_p2bs

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. b : pv11_p1_Ballot_Num()@i
19. Inj(Id;ℤ;ldrs_uid)@i
20. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
21. e : E@i
22. ∀e1:E
      ((e1 < e)
      ⇒ (∀e0:E. ∀s:ℤ. ∀c:Cmd. ∀waitfor:bag(Id). ∀i,l:Id.
            (start ≤loc e1
            ⇒ pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e0)
            ⇒ (waitfor = pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es.start;e1) ∈ bag(Id))
            ⇒ (↓∃L:E List
                  ((waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id))
                  ∧ (∀e':E
                       ((e' ∈ L)
                       ⇒ ((e' < e1)
                          ∧ loc(e') ↓∈ accpts

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

                          ∧ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))))))))))

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

∧ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))))))))
BY

{ (RepUR ``pv11_p1_on_p2b`` 0 THEN Repeat (AutoPairEta [1;3;1] 0) THEN Repeat (AutoSplit) THEN (D 0 THENA 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

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

                          ∧ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))))))))))

23. e0 : E@i
24. s : ℤ@i
25. c : Cmd@i
26. waitfor : bag(Id)@i
27. i : Id@i
28. l : Id@i
29. start ≤loc e @i
30. pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e0)@i
31. ↑e ∈_b pv11_p1_p2b'base(Cmd;f)
32. ↑first(e)
33. header(e) = ``pv11_p1 p2b`` ∈ Name
34. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
35. b = (fst(snd(msgval(e)))) ∈ pv11_p1_Ballot_Num()
36. s = (fst(snd(snd(msgval(e))))) ∈ ℤ
37. b = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num()
38. waitfor = accpts - fst(msgval(e)) ∈ bag(Id)@i
⊢ ↓∃L:E List
    ((waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id))
    ∧ (∀e':E
         ((e' ∈ L)
         ⇒ ((e' < e)
            ∧ loc(e') ↓∈ accpts
            ∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())
            ∧ (<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

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. b : pv11_p1_Ballot_Num()@i
19. Inj(Id;ℤ;ldrs_uid)@i
20. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
21. e : E@i
22. ¬(b = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())
23. ∀e1:E
      ((e1 < e)
      ⇒ (∀e0:E. ∀s:ℤ. ∀c:Cmd. ∀waitfor:bag(Id). ∀i,l:Id.
            (start ≤loc e1
            ⇒ pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e0)
            ⇒ (waitfor = pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es.start;e1) ∈ bag(Id))
            ⇒ (↓∃L:E List
                  ((waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id))
                  ∧ (∀e':E
                       ((e' ∈ L)
                       ⇒ ((e' < e1)
                          ∧ loc(e') ↓∈ accpts

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

                          ∧ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))))))))))

24. e0 : E@i
25. s : ℤ@i
26. c : Cmd@i
27. waitfor : bag(Id)@i
28. i : Id@i
29. l : Id@i
30. start ≤loc e @i
31. pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e0)@i
32. ↑e ∈_b pv11_p1_p2b'base(Cmd;f)
33. ↑first(e)
34. header(e) = ``pv11_p1 p2b`` ∈ Name
35. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
36. b = (fst(snd(msgval(e)))) ∈ pv11_p1_Ballot_Num()
37. s = (fst(snd(snd(msgval(e))))) ∈ ℤ
38. waitfor = accpts ∈ bag(Id)@i
⊢ ↓∃L:E List
    ((waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id))
    ∧ (∀e':E
         ((e' ∈ L)
         ⇒ ((e' < e)
            ∧ loc(e') ↓∈ accpts
            ∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())
            ∧ (<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

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

                          ∧ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))))))))))

23. e0 : E@i
24. s : ℤ@i
25. s ≠ fst(snd(snd(msgval(e))))
26. c : Cmd@i
27. waitfor : bag(Id)@i
28. i : Id@i
29. l : Id@i
30. start ≤loc e @i
31. pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e0)@i
32. ↑e ∈_b pv11_p1_p2b'base(Cmd;f)
33. ↑first(e)
34. header(e) = ``pv11_p1 p2b`` ∈ Name
35. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
36. b = (fst(snd(msgval(e)))) ∈ pv11_p1_Ballot_Num()
37. waitfor = accpts ∈ bag(Id)@i
⊢ ↓∃L:E List
    ((waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id))
    ∧ (∀e':E
         ((e' ∈ L)
         ⇒ ((e' < e)
            ∧ loc(e') ↓∈ accpts
            ∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())
            ∧ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))))))

4
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. b : pv11_p1_Ballot_Num()@i
19. Inj(Id;ℤ;ldrs_uid)@i
20. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
21. e : E@i
22. ¬(b = (fst(snd(msgval(e)))) ∈ pv11_p1_Ballot_Num())
23. ∀e1:E
      ((e1 < e)
      ⇒ (∀e0:E. ∀s:ℤ. ∀c:Cmd. ∀waitfor:bag(Id). ∀i,l:Id.
            (start ≤loc e1
            ⇒ pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e0)
            ⇒ (waitfor = pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es.start;e1) ∈ bag(Id))
            ⇒ (↓∃L:E List
                  ((waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id))
                  ∧ (∀e':E
                       ((e' ∈ L)
                       ⇒ ((e' < e1)
                          ∧ loc(e') ↓∈ accpts

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

                          ∧ (<b, s, c> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))))))))))

24. e0 : E@i
25. s : ℤ@i
26. c : Cmd@i
27. waitfor : bag(Id)@i
28. i : Id@i
29. l : Id@i
30. start ≤loc e @i
31. pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e0)@i
32. ↑e ∈_b pv11_p1_p2b'base(Cmd;f)
33. ↑first(e)
34. header(e) = ``pv11_p1 p2b`` ∈ Name
35. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
36. waitfor = accpts ∈ bag(Id)@i
⊢ ↓∃L:E List
    ((waitfor = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id))
    ∧ (∀e':E
         ((e' ∈ L)
         ⇒ ((e' < e)
            ∧ loc(e') ↓∈ accpts
            ∧ (b = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num())
            ∧ (<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. start : E@i 14. accpts : bag(Id)@i 15. ldrs : bag(Id)@i 16. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 17. reps : bag(Id)@i 18. b : pv11\_p1\_Ballot\_Num()@i 19. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 20. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 21. e : E@i 22. \mforall{}e1:E ((e1 < e) {}\mRightarrow{} (\mforall{}e0:E. \mforall{}s:\mBbbZ{}. \mforall{}c:Cmd. \mforall{}waitfor:bag(Id). \mforall{}i,l:Id. (start \mleq{}loc e1 {}\mRightarrow{} pv11\_p1\_p2a'send(Cmd;f) i <l, b, s, c> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_\mbackslash{}\000Cff7buid}$;reps;f)(e0) {}\mRightarrow{} (waitfor = pv11\_p1\_CommanderStateFun(Cmd;accpts;f;b;s;es.start;e1)) {}\mRightarrow{} (\mdownarrow{}\mexists{}L:E List ((waitfor = [i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))]) \mwedge{} (\mforall{}e':E ((e' \mmember{} L) {}\mRightarrow{} ((e' < e1) \mwedge{} loc(e') \mdownarrow{}\mmember{} accpts \mwedge{} (b = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;\000Cf;es;e')))) \mwedge{} (<b, s, c> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid\mbackslash{}ff7\000Cd$;f;es;e'))))))))))) 23. e0 : E@i 24. s : \mBbbZ{}@i 25. c : Cmd@i 26. waitfor : bag(Id)@i 27. i : Id@i 28. l : Id@i 29. start \mleq{}loc e @i 30. pv11\_p1\_p2a'send(Cmd;f) i <l, b, s, c> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid\mbackslash{}ff7\000Cd$;reps;f)(e0)@i 31. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2b'base(Cmd;f) 32. \muparrow{}first(e) 33. header(e) = ``pv11\_p1 p2b`` 34. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num()) \mvdash{} (waitfor = (pv11\_p1\_on\_p2b() b s loc(e) msgval(e) accpts)) {}\mRightarrow{} (\mdownarrow{}\mexists{}L:E List ((waitfor = [i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))]) \mwedge{} (\mforall{}e':E ((e' \mmember{} L) {}\mRightarrow{} ((e' < e) \mwedge{} loc(e') \mdownarrow{}\mmember{} accpts \mwedge{} (b = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e')))) \mwedge{} (<b, s, c> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;\000Ce')))))))) By Latex: (RepUR ``pv11\_p1\_on\_p2b`` 0 THEN Repeat (AutoPairEta [1;3;1] 0) THEN Repeat (AutoSplit) THEN (D 0 THENA Auto)) Home Index