Step
*
1
3
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. ¬↑e ∈b pv11_p1_p2b'base(Cmd;f)
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. ↑first(e)
⊢ (waitfor = accpts ∈ 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
{ ((D 0 THENA Auto) THEN D 0 THEN InstConcl [⌈[]⌉]⋅ THEN Reduce 0 THEN Auto) }
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{}e \mmember{}\msubb{} pv11\_p1\_p2b'base(Cmd;f)
23. \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')))))))))))
24. e0 : E@i
25. s : \mBbbZ{}@i
26. c : Cmd@i
27. waitfor : bag(Id)@i
28. i : Id@i
29. l : Id@i
30. start \mleq{}loc e @i
31. 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
32. \muparrow{}first(e)
\mvdash{} (waitfor = 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:
((D 0 THENA Auto) THEN D 0 THEN InstConcl [\mkleeneopen{}[]\mkleeneclose{}]\mcdot{} THEN Reduce 0 THEN Auto)
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