Proof of Nuprl Lemma : pv11_p1_commander_state_from

Step * 1 4 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. ¬↑first(e)
23. ¬↑e ∈_b pv11_p1_p2b'base(Cmd;f)
24. ∀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')))))))))))

25. e0 : E@i
26. s : ℤ@i
27. c : Cmd@i
28. waitfor : bag(Id)@i
29. i : Id@i
30. l : Id@i
31. start ≤loc e @i
32. pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e0)@i
⊢ (waitfor = pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es.start;pred(e)) ∈ bag(Id))
⇒ (↓∃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'))))))))
BY
{ ((Assert ⌈¬↑first(e)⌉⋅ THENA Auto)
THEN (Assert ⌈(start <loc e)⌉⋅

         THENA (Auto THEN D (-12) THEN RWO "eo-forward-first" 0 THEN Auto THEN AutoSplit THEN D (-12) THEN Auto)

)

   THEN (InstHyp [⌈pred(e)⌉;⌈e0⌉;⌈s⌉;⌈c⌉;⌈pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es.start;pred(e))⌉;⌈i⌉;⌈l⌉] (-11)⋅

         THENA Auto
         )
   THEN (Assert ⌈start ≤loc pred(e) ⌉⋅ THENA (RWO "eo-forward-pred" 0⋅ THEN Auto))
   THEN (Subst ⌈pred(e) = pred(e) ∈ E⌉ 0⋅
         THENA (Try (RWO "equal-eo-forward-E" 0) THEN Try (BLemma `eo-forward-pred`) THEN Auto)
         )
   THEN (SqExRepD 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')))))))))))

36. pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es.start;pred(e)) = [i∈accpts|¬_bi ∈_b map(λe.loc(e);L))] ∈ bag(Id)

37. ∀e':E
      ((e' ∈ L)
      ⇒ ((e' < pred(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')))))
38. start ≤loc pred(e)
39. waitfor = pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es.start;pred(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')))))))

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. \mneg{}\muparrow{}first(e) 23. \mneg{}\muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2b'base(Cmd;f) 24. \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'))))))))))) 25. e0 : E@i 26. s : \mBbbZ{}@i 27. c : Cmd@i 28. waitfor : bag(Id)@i 29. i : Id@i 30. l : Id@i 31. start \mleq{}loc e @i 32. 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 \mvdash{} (waitfor = pv11\_p1\_CommanderStateFun(Cmd;accpts;f;b;s;es.start;pred(e))) {}\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: ((Assert \mkleeneopen{}\mneg{}\muparrow{}first(e)\mkleeneclose{}\mcdot{} THENA Auto) THEN (Assert \mkleeneopen{}(start <loc e)\mkleeneclose{}\mcdot{} THENA (Auto THEN D (-12) THEN RWO "eo-forward-first" 0 THEN Auto THEN AutoSplit THEN D (-12) THEN Auto) ) THEN (InstHyp [\mkleeneopen{}pred(e)\mkleeneclose{};\mkleeneopen{}e0\mkleeneclose{};\mkleeneopen{}s\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{}; \mkleeneopen{}pv11\_p1\_CommanderStateFun(Cmd;accpts;f;b;s;es.start;pred(e))\mkleeneclose{};\mkleeneopen{}i\mkleeneclose{};\mkleeneopen{}l\mkleeneclose{}] (-11)\mcdot{} THENA Auto ) THEN (Assert \mkleeneopen{}start \mleq{}loc pred(e) \mkleeneclose{}\mcdot{} THENA (RWO "eo-forward-pred" 0\mcdot{} THEN Auto)) THEN (Subst \mkleeneopen{}pred(e) = pred(e)\mkleeneclose{} 0\mcdot{} THENA (Try (RWO "equal-eo-forward-E" 0) THEN Try (BLemma `eo-forward-pred`) THEN Auto) ) THEN (SqExRepD THENA Auto)) Home Index