Step
*
1
4
1
1
1
1
of Lemma
pv11_p1_validity-lemma
1. Cmd : {T:Type| valueall-type(T)}
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. ∀e1:E
((e1 < e)
⇒ (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e1;f)
∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e1;f)
∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.
(start ≤loc e1 ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
∧ ((header(e1) = ``pv11_p1 adopted`` ∈ Name)
⇒ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-adopted-message(Cmd;es;e1;f))
∧ ((header(e1) = ``pv11_p1 p1b`` ∈ Name)
⇒ has-es-info-type(es;e1;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-p1b-message(Cmd;es;e1;f))
∧ ((header(e1) = ``pv11_p1 p2a`` ∈ Name)
⇒ has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
⇒ pv11_p1_valid-p2a-message(Cmd;es;e1;f))))
20. pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
21. pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
22. ∀b:pv11_p1_Ballot_Num(). ∀start:E. (start ≤loc e ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f))
23. header(e) = ``pv11_p1 adopted`` ∈ Name@i
24. has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
25. p1 : ℤ@i
26. p2 : Cmd@i
27. (<p1, p2> ∈ map(λp.(snd(p));snd(msgval(e))))@i
28. ∀e:E
((header(e) ∈ pv11_p1_headers_no_inputs())
⇒ (↓∃e':E
∃delay:ℤ
((e' < e) ∧ make-msg-interface(delay;loc(e);info(e)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e'))))
29. e' : E
30. delay : ℤ
31. (e' < e)
32. loc(e') ↓∈ ldrs
33. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e')) o pv11_p1_p1b'base(Cmd;f)) o
pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_init_ballot_num() loc(e'))) prior to e')
34. es-init(es;e') ≤loc e'
35. header(e') = ``pv11_p1 p1b`` ∈ Name
36. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
37. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
38. delay = 0 ∈ ℤ
39. loc(e) = loc(e') ∈ Id
40. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
41. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
42. es-info-auth(es;e) = pv11_p1_init_active()
43. (fst(msgval(e))) = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num()
44. (snd(msgval(e)))
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
45. pv11_p1_valid-ScoutState(Cmd;accpts;pv11_p1_init_ballot_num() loc(e');es-init(es;e');es;e';f)
⊢ ↓pv11_p1_valid-proposal(Cmd;es;e;<p1, p2>;f)
BY
{ (D 0
THEN Using [`e',⌈e'⌉] (BLemma `pv11_p1_valid-proposal-transitivity`)⋅
THEN Auto
THEN RepUR ``pv11_p1_valid-ScoutState pv11_p1_valid-proposals`` (-1)
THEN (RWO "l_all_iff" (-1) THENA Auto)
THEN (With ⌈<p1, p2>⌉ (D (-1))⋅ THENA Auto)
THEN (RWO "eo-forward-trivial" (-1) THEN Auto)
THEN RWO "es-first-init" 0
THEN Auto) }
Latex:
Latex:
1. Cmd : \{T:Type| valueall-type(T)\}
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. \mforall{}e1:E
((e1 < e)
{}\mRightarrow{} (pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e1;f)
\mwedge{} pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e1;f)
\mwedge{} (\mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}start:E.
(start \mleq{}loc e1 {}\mRightarrow{} pv11\_p1\_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
\mwedge{} ((header(e1) = ``pv11\_p1 adopted``)
{}\mRightarrow{} has-es-info-type(es;e1;f;pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
{}\mRightarrow{} pv11\_p1\_valid-adopted-message(Cmd;es;e1;f))
\mwedge{} ((header(e1) = ``pv11\_p1 p1b``)
{}\mRightarrow{} has-es-info-type(es;e1;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;e1;f))
\mwedge{} ((header(e1) = ``pv11\_p1 p2a``)
{}\mRightarrow{} has-es-info-type(es;e1;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
{}\mRightarrow{} pv11\_p1\_valid-p2a-message(Cmd;es;e1;f))))
20. pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e;f)
21. pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e;f)
22. \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))
23. header(e) = ``pv11\_p1 adopted``@i
24. has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))@i
25. p1 : \mBbbZ{}@i
26. p2 : Cmd@i
27. (<p1, p2> \mmember{} map(\mlambda{}p.(snd(p));snd(msgval(e))))@i
28. \mforall{}e:E
((header(e) \mmember{} pv11\_p1\_headers\_no\_inputs())
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
\mexists{}delay:\mBbbZ{}
((e' < e)
\mwedge{} make-msg-interface(delay;loc(e);info(e)) \mmember{}
pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}$;reps;f)(e'))))
29. e' : E
30. delay : \mBbbZ{}
31. (e' < e)
32. loc(e') \mdownarrow{}\mmember{} ldrs
33. (no ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_init\_ballot\_num() loc(e')) o
pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f)
(pv11\_p1\_init\_ballot\_num() loc(e'))) prior to e')
34. es-init(es;e') \mleq{}loc e'
35. header(e') = ``pv11\_p1 p1b``
36. 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))
37. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(msgval(e'))))
38. delay = 0
39. loc(e) = loc(e')
40. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(snd(msgval(e')))))
41. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num()
loc(e');es;e'))) < pv11\_p1\_threshold(accpts)
42. es-info-auth(es;e) = pv11\_p1\_init\_active()
43. (fst(msgval(e))) = (pv11\_p1\_init\_ballot\_num() loc(e'))
44. (snd(msgval(e)))
= (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e')))
45. pv11\_p1\_valid-ScoutState(Cmd;accpts;pv11\_p1\_init\_ballot\_num() loc(e');es-init(es;e');es;e';f)
\mvdash{} \mdownarrow{}pv11\_p1\_valid-proposal(Cmd;es;e;<p1, p2>f)
By
Latex:
(D 0
THEN Using [`e',\mkleeneopen{}e'\mkleeneclose{}] (BLemma `pv11\_p1\_valid-proposal-transitivity`)\mcdot{}
THEN Auto
THEN RepUR ``pv11\_p1\_valid-ScoutState pv11\_p1\_valid-proposals`` (-1)
THEN (RWO "l\_all\_iff" (-1) THENA Auto)
THEN (With \mkleeneopen{}<p1, p2>\mkleeneclose{} (D (-1))\mcdot{} THENA Auto)
THEN (RWO "eo-forward-trivial" (-1) THEN Auto)
THEN RWO "es-first-init" 0
THEN Auto)
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