Step
*
2
of Lemma
pv11_p1_adopted_from_acceptor
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. e : E@i
14. accpts : bag(Id)@i
15. ldrs : bag(Id)@i
16. ldrs_uid : Id ⟶ ℤ@i
17. reps : bag(Id)@i
18. bnum : pv11_p1_Ballot_Num()@i
19. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
20. b : pv11_p1_Ballot_Num()@i
21. s : ℤ@i
22. c : Cmd@i
23. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
24. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)@i
25. (<b, s, c> ∈ pvals)@i
26. e' : E
27. delay : ℤ
28. (e' < e)
29. loc(e') ↓∈ ldrs
30. e1 : {e1:E| e1 ≤loc e' }
31. header(e1) = ``pv11_p1 preempted`` ∈ Name
32. has-es-info-type(es;e1;f;pv11_p1_Ballot_Num())
33. ↑(pv11_p1_is_bnum() msgval(e1))
34. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)) < msgval(e1))
35. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1)) o pv11_p1_p1b'base(Cmd;f)) o
pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1))) between e1 and e')
36. header(e') = ``pv11_p1 p1b`` ∈ Name
37. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
38. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
39. delay = 0 ∈ ℤ
40. loc(e) = loc(e') ∈ Id
41. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
42. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1)
loc(e1);es.e1;e'))) < pv11_p1_threshold(accpts)
43. msg-authentic(info(e)) = pv11_p1_init_active()
44. bnum = (pv11_p1_upd_bnum() msgval(e1) loc(e1)) ∈ pv11_p1_Ballot_Num()
45. pvals
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1) loc(e1);es.e1;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
⊢ ↓∃e':E
∃accepted:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
((e' < e) ∧ <bnum, accepted> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e') ∧ (<b, s, c> ∈ accepted))
BY
{ ((InstLemma `pv11_p1_scout_from_acc` [⌜Cmd⌝;⌜f⌝;⌜es.e1⌝;⌜e'⌝;⌜accpts⌝;
⌜pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1) loc(e1);es.e1;e')⌝;⌜pv11_p1_upd_bnum() msgval(e1)
loc(e1)⌝]⋅
THENA (Auto THEN BLemma `pv11_p1_ScoutState-classrel` THEN Auto)
)
THEN UsePairEta [1] (-1)
THEN Reduce (-1)
THEN (RWO "-2<" (-1) THENA Auto)
THEN (InstHyp [⌜<b, s, c>⌝] (-1)⋅ THENA Auto)
THEN ExRepD
THEN RenameVar `e2' (-6)) }
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
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. e : E@i
14. accpts : bag(Id)@i
15. ldrs : bag(Id)@i
16. ldrs_uid : Id ⟶ ℤ@i
17. reps : bag(Id)@i
18. bnum : pv11_p1_Ballot_Num()@i
19. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
20. b : pv11_p1_Ballot_Num()@i
21. s : ℤ@i
22. c : Cmd@i
23. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
24. <bnum, pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e)@i
25. (<b, s, c> ∈ pvals)@i
26. e' : E
27. delay : ℤ
28. (e' < e)
29. loc(e') ↓∈ ldrs
30. e1 : {e1:E| e1 ≤loc e' }
31. header(e1) = ``pv11_p1 preempted`` ∈ Name
32. has-es-info-type(es;e1;f;pv11_p1_Ballot_Num())
33. ↑(pv11_p1_is_bnum() msgval(e1))
34. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)) < msgval(e1))
35. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1)) o pv11_p1_p1b'base(Cmd;f)) o
pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e1) loc(e1))) between e1 and e')
36. header(e') = ``pv11_p1 p1b`` ∈ Name
37. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
38. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
39. delay = 0 ∈ ℤ
40. loc(e) = loc(e') ∈ Id
41. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
42. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1)
loc(e1);es.e1;e'))) < pv11_p1_threshold(accpts)
43. msg-authentic(info(e)) = pv11_p1_init_active()
44. bnum = (pv11_p1_upd_bnum() msgval(e1) loc(e1)) ∈ pv11_p1_Ballot_Num()
45. pvals
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1) loc(e1);es.e1;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
46. ∀p:pv11_p1_Ballot_Num() × ℤ × Cmd
((p ∈ pvals)
⇒ (∃e'@0:E
∃l:Id
∃r:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
(e'@0 ≤loc e'
∧ <l, pv11_p1_upd_bnum() msgval(e1) loc(e1), pv11_p1_upd_bnum() msgval(e1) loc(e1), r> ∈
pv11_p1_p1b'base(Cmd;f)(e'@0)
∧ (p ∈ r))))
47. e2 : E
48. l : Id
49. r : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
50. e2 ≤loc e'
51. <l, pv11_p1_upd_bnum() msgval(e1) loc(e1), pv11_p1_upd_bnum() msgval(e1) loc(e1), r> ∈ pv11_p1_p1b'base(Cmd;f)(e2)
52. (<b, s, c> ∈ r)
⊢ ↓∃e':E
∃accepted:(pv11_p1_Ballot_Num() × ℤ × Cmd) List
((e' < e) ∧ <bnum, accepted> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e') ∧ (<b, s, c> ∈ accepted))
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. e : 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. bnum : pv11\_p1\_Ballot\_Num()@i
19. pvals : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
20. b : pv11\_p1\_Ballot\_Num()@i
21. s : \mBbbZ{}@i
22. c : Cmd@i
23. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
24. <bnum, pvals> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e)@i
25. (<b, s, c> \mmember{} pvals)@i
26. e' : E
27. delay : \mBbbZ{}
28. (e' < e)
29. loc(e') \mdownarrow{}\mmember{} ldrs
30. e1 : \{e1:E| e1 \mleq{}loc e' \}
31. header(e1) = ``pv11\_p1 preempted``
32. has-es-info-type(es;e1;f;pv11\_p1\_Ballot\_Num())
33. \muparrow{}(pv11\_p1\_is\_bnum() msgval(e1))
34. \muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)) < msgval(e1))
35. (no ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) o
pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f)
(pv11\_p1\_upd\_bnum() msgval(e1) loc(e1))) between e1 and e')
36. header(e') = ``pv11\_p1 p1b``
37. 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))
38. (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e'))))
39. delay = 0
40. loc(e) = loc(e')
41. (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e')))))
42. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1)
loc(e1);es.e1;e'))) < pv11\_p1\_threshold(accpts)
43. msg-authentic(info(e)) = pv11\_p1\_init\_active()
44. bnum = (pv11\_p1\_upd\_bnum() msgval(e1) loc(e1))
45. pvals
= (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1) loc(e1);es.e1;e')))
\mvdash{} \mdownarrow{}\mexists{}e':E
\mexists{}accepted:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
((e' < e)
\mwedge{} <bnum, accepted> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e')
\mwedge{} (<b, s, c> \mmember{} accepted))
By
Latex:
((InstLemma `pv11\_p1\_scout\_from\_acc` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es.e1\mkleeneclose{};\mkleeneopen{}e'\mkleeneclose{};\mkleeneopen{}accpts\mkleeneclose{};
\mkleeneopen{}pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e1) loc(e1);es.e1;e')\mkleeneclose{};
\mkleeneopen{}pv11\_p1\_upd\_bnum() msgval(e1) loc(e1)\mkleeneclose{}]\mcdot{}
THENA (Auto THEN BLemma `pv11\_p1\_ScoutState-classrel` THEN Auto)
)
THEN UsePairEta [1] (-1)
THEN Reduce (-1)
THEN (RWO "-2<" (-1) THENA Auto)
THEN (InstHyp [\mkleeneopen{}<b, s, c>\mkleeneclose{}] (-1)\mcdot{} THENA Auto)
THEN ExRepD
THEN RenameVar `e2' (-6))
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