Step
*
1
2
1
of Lemma
pv11_p1-validity
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. i : Id@i
20. p1 : ℤ@i
21. p2 : Cmd@i
22. loc(e) ↓∈ ldrs
23. e' : {e':E| e' ≤loc e }
24. header(e) = ``pv11_p1 p2b`` ∈ Name
25. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
26. 0 = 0 ∈ ℤ
27. (fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num()
28. #(pv11_p1_CommanderStateFun(Cmd;accpts;f;fst(snd(msgval(e)));fst(snd(snd(msgval(e))));es.e';e)) < ...
29. i ↓∈ reps
30. ff = pv11_p1_init_active()
31. p1 = (fst(snd(snd(msgval(e))))) ∈ ℤ
32. header(e') = ``pv11_p1 adopted`` ∈ Name
33. has-es-info-type(es;e';f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
34. (fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
35. <fst(snd(snd(msgval(e)))), p2> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))
36. ¬(∃p2:Cmd. (<fst(snd(snd(msgval(e)))), p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e')))))
37. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()
38. ∀e:E
(pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E. (start ≤loc e ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f)))
∧ ((header(e) = ``pv11_p1 adopted`` ∈ Name)
⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f))
∧ ((header(e) = ``pv11_p1 p1b`` ∈ Name)
⇒ has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-p1b-message(Cmd;es;e;f))
∧ ((header(e) = ``pv11_p1 p2a`` ∈ Name)
⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
⇒ pv11_p1_valid-p2a-message(Cmd;es;e;f)))
39. <p1, p2> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))
⊢ pv11_p1_valid-proposal(Cmd;es;e;<p1, p2>;f)
BY
{ ((FLemma `bag-member-sq-list-member` [-1] THENA Auto)
THEN SqExRepD
THEN (Using [`e',⌈e'⌉] (BLemma `pv11_p1_valid-proposal-transitivity`)⋅
THEN Auto
THEN Try (Complete ((DSetVars THEN Unhide THEN Auto))))
THEN Assert ⌈pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e';f)⌉⋅
THEN Auto
THEN RepUR ``pv11_p1_valid-LeaderState pv11_p1_valid-proposals`` (-1)
THEN (RWO "l_all_iff" (-1) THENA Auto)
THEN BHyp (-1)
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
16. reps : bag(Id)@i
17. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
18. e : E@i
19. i : Id@i
20. p1 : \mBbbZ{}@i
21. p2 : Cmd@i
22. loc(e) \mdownarrow{}\mmember{} ldrs
23. e' : \{e':E| e' \mleq{}loc e \}
24. header(e) = ``pv11\_p1 p2b``
25. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num())
26. 0 = 0
27. (fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e)))))
28. \#(pv11\_p1\_CommanderStateFun(Cmd;accpts;f;fst(snd(msgval(e)));fst(snd(snd(...)));es.e';e)) < ...
29. i \mdownarrow{}\mmember{} reps
30. ff = pv11\_p1\_init\_active()
31. p1 = (fst(snd(snd(msgval(e)))))
32. header(e') = ``pv11\_p1 adopted``
33. has-es-info-type(es;e';f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
34. (fst(msgval(e'))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e')))
35. <fst(snd(snd(msgval(e)))), p2> \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}\000C$;f;es;e')))
36. \mneg{}(\mexists{}p2:Cmd. (<fst(snd(snd(msgval(e)))), p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) \000C(snd(msgval(e')))))
37. (fst(snd(msgval(e)))) = (fst(msgval(e')))
38. \mforall{}e:E
(pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e;f)
\mwedge{} pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e;f)
\mwedge{} (\mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}start:E.
(start \mleq{}loc e {}\mRightarrow{} pv11\_p1\_valid-ScoutState(Cmd;accpts;b;start;es;e;f)))
\mwedge{} ((header(e) = ``pv11\_p1 adopted``)
{}\mRightarrow{} has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
{}\mRightarrow{} pv11\_p1\_valid-adopted-message(Cmd;es;e;f))
\mwedge{} ((header(e) = ``pv11\_p1 p1b``)
{}\mRightarrow{} has-es-info-type(es;e;f;Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
{}\mRightarrow{} pv11\_p1\_valid-p1b-message(Cmd;es;e;f))
\mwedge{} ((header(e) = ``pv11\_p1 p2a``)
{}\mRightarrow{} has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
{}\mRightarrow{} pv11\_p1\_valid-p2a-message(Cmd;es;e;f)))
39. <p1, p2> \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e')))
\mvdash{} pv11\_p1\_valid-proposal(Cmd;es;e;<p1, p2>f)
By
Latex:
((FLemma `bag-member-sq-list-member` [-1] THENA Auto)
THEN SqExRepD
THEN (Using [`e',\mkleeneopen{}e'\mkleeneclose{}] (BLemma `pv11\_p1\_valid-proposal-transitivity`)\mcdot{}
THEN Auto
THEN Try (Complete ((DSetVars THEN Unhide THEN Auto))))
THEN Assert \mkleeneopen{}pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e';f)\mkleeneclose{}\mcdot{}
THEN Auto
THEN RepUR ``pv11\_p1\_valid-LeaderState pv11\_p1\_valid-proposals`` (-1)
THEN (RWO "l\_all\_iff" (-1) THENA Auto)
THEN BHyp (-1)
THEN Auto)
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