Step
*
1
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') = [propose] ∈ Name
33. has-es-info-type(es;e';f;ℤ × Cmd)
34. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))))
35. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e'))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))))
36. (fst(snd(msgval(e)))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
37. <fst(snd(snd(msgval(e)))), p2> = msgval(e') ∈ (ℤ × Cmd)
38. (no ((pv11_p1_commander_output(Cmd;accpts;reps;f) <fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), p2> o pv11_p1_p2b'\000Cbase(Cmd;f)) o
pv11_p1_CommanderState(Cmd;accpts;f) (fst(snd(msgval(e)))) (fst(snd(snd(msgval(e)))))) between e' and e)
39. ∀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)))
⊢ pv11_p1_valid-proposal(Cmd;es;e;<p1, p2>;f)
BY
{ (D 0
THEN Auto
THEN With ⌈e'⌉ (D 0)⋅
THEN Auto
THEN Try (Complete ((DSetVars THEN Unhide THEN Auto)))
THEN RepeatFor 2 ((RW ClassRelStepC 0 THEN Auto))
THEN D 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. 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') = [propose]
33. has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} Cmd)
34. \muparrow{}(fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e'))))
35. \mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) (fst(msgval(e')))
(snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e')))))
36. (fst(snd(msgval(e)))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e'\000C)))
37. <fst(snd(snd(msgval(e)))), p2> = msgval(e')
38. (no ((pv11\_p1\_commander\_output(Cmd;accpts;reps;f)
<fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), p2> o pv11\_p1\_p2b'base(Cmd;f)) o
pv11\_p1\_CommanderState(Cmd;accpts;f) (fst(snd(msgval(e)))) (fst(snd(snd(msgval(e))))))
between e' and e)
39. \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)))
\mvdash{} pv11\_p1\_valid-proposal(Cmd;es;e;<p1, p2>f)
By
Latex:
(D 0
THEN Auto
THEN With \mkleeneopen{}e'\mkleeneclose{} (D 0)\mcdot{}
THEN Auto
THEN Try (Complete ((DSetVars THEN Unhide THEN Auto)))
THEN RepeatFor 2 ((RW ClassRelStepC 0 THEN Auto))
THEN D 0
THEN Auto)
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