Step
*
1
2
1
1
of Lemma
pv11_p1_scout_state_subset_pvals
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. ¬↑first(e)
15. bnum : pv11_p1_Ballot_Num()@i
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ─→ ℤ@i
19. reps : bag(Id)@i
20. Inj(Id;ℤ;ldrs_uid)@i
21. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
22. header(e) = ``pv11_p1 p1b`` ∈ Name@i
23. has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
24. (fst(snd(msgval(e)))) = bnum ∈ pv11_p1_Ballot_Num()@i
25. (fst(snd(snd(msgval(e))))) = bnum ∈ pv11_p1_Ballot_Num()@i
26. ↑e ∈b pv11_p1_p1b'base(Cmd;f)
27. msgval(e) ∈ Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
28. v1 : Id@i
29. v3 : pv11_p1_Ballot_Num()@i
30. v5 : pv11_p1_Ballot_Num()@i
31. v6 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
32. msgval(e)
= <v1, v3, v5, v6>
∈ (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
33. bnum = v3 ∈ pv11_p1_Ballot_Num()
34. bnum = v5 ∈ pv11_p1_Ballot_Num()
35. ∀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'))))
36. e' : E
37. delay : ℤ
38. (e' < e)
39. loc(e') ↓∈ accpts
40. header(e') = ``pv11_p1 p1a`` ∈ Name
41. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num())
42. delay = 0 ∈ ℤ
43. loc(e) = (fst(msgval(e'))) ∈ Id
44. es-info-auth(es;e) = pv11_p1_init_active()
45. (fst(msgval(e))) = loc(e') ∈ Id
46. (fst(snd(msgval(e)))) = (snd(msgval(e'))) ∈ pv11_p1_Ballot_Num()
47. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))>
= pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⊢ ↓l_subset(pv11_p1_Ballot_Num() × ℤ × Cmd;v6;pv11_p1_append_news(Cmd) pv11_p1_same_pvalue(Cmd)
(snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es;pred(e))))
v6)
BY
{ (RepeatFor 3 (SimpEqPairs)
THEN D 0
THEN Unfold `l_subset` 0
THEN Auto
THEN (InstLemma `pv11_p1_append_news_iff2` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈accpts⌉;⌈ldrs⌉;⌈ldrs_uid⌉;⌈reps⌉;⌈x⌉;
⌈snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es;pred(e)))⌉;⌈v6⌉]⋅
THENM (RWO "-1" 0 THEN OrRight THEN Auto)
)
THEN Auto
THEN RWO "l_all_iff" 0
THEN Auto
THEN GenListD (-1)
THEN DProdsAndUnions
THEN Auto) }
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. ¬↑first(e)
15. bnum : pv11_p1_Ballot_Num()@i
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ─→ ℤ@i
19. reps : bag(Id)@i
20. Inj(Id;ℤ;ldrs_uid)@i
21. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
22. header(e) = ``pv11_p1 p1b`` ∈ Name@i
23. has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
24. (fst(snd(msgval(e)))) = bnum ∈ pv11_p1_Ballot_Num()@i
25. (fst(snd(snd(msgval(e))))) = bnum ∈ pv11_p1_Ballot_Num()@i
26. ↑e ∈b pv11_p1_p1b'base(Cmd;f)
27. msgval(e) ∈ Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
28. v1 : Id@i
29. v3 : pv11_p1_Ballot_Num()@i
30. v5 : pv11_p1_Ballot_Num()@i
31. v6 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
32. (fst(msgval(e))) = v1 ∈ Id
33. (fst(snd(msgval(e)))) = v3 ∈ pv11_p1_Ballot_Num()
34. (fst(snd(snd(msgval(e))))) = v5 ∈ pv11_p1_Ballot_Num()
35. (snd(snd(snd(msgval(e))))) = v6 ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
36. bnum = v3 ∈ pv11_p1_Ballot_Num()
37. bnum = v5 ∈ pv11_p1_Ballot_Num()
38. ∀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'))))
39. e' : E
40. delay : ℤ
41. (e' < e)
42. loc(e') ↓∈ accpts
43. header(e') = ``pv11_p1 p1a`` ∈ Name
44. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num())
45. delay = 0 ∈ ℤ
46. loc(e) = (fst(msgval(e'))) ∈ Id
47. es-info-auth(es;e) = pv11_p1_init_active()
48. (fst(msgval(e))) = loc(e') ∈ Id
49. (fst(snd(msgval(e)))) = (snd(msgval(e'))) ∈ pv11_p1_Ballot_Num()
50. (fst(snd(snd(msgval(e))))) = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
51. (snd(snd(snd(msgval(e)))))
= (snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
52. x4 : pv11_p1_Ballot_Num()@i
53. x8 : ℤ@i
54. x9 : Cmd@i
55. (<x4, x8, x9> ∈ v6)@i
56. x2 : pv11_p1_Ballot_Num()@i
57. x6 : ℤ@i
58. x7 : Cmd@i
59. (<x2, x6, x7> ∈ snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;bnum;es;pred(e))))
⊢ ↓∃e:E. (<x2, x6, x7> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)))
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. \mneg{}\muparrow{}first(e)
15. bnum : pv11\_p1\_Ballot\_Num()@i
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
19. reps : bag(Id)@i
20. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
21. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
22. header(e) = ``pv11\_p1 p1b``@i
23. 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))@i
24. (fst(snd(msgval(e)))) = bnum@i
25. (fst(snd(snd(msgval(e))))) = bnum@i
26. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p1b'base(Cmd;f)
27. msgval(e) \mmember{} Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)
28. v1 : Id@i
29. v3 : pv11\_p1\_Ballot\_Num()@i
30. v5 : pv11\_p1\_Ballot\_Num()@i
31. v6 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
32. msgval(e) = <v1, v3, v5, v6>@i
33. bnum = v3
34. bnum = v5
35. \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'))))
36. e' : E
37. delay : \mBbbZ{}
38. (e' < e)
39. loc(e') \mdownarrow{}\mmember{} accpts
40. header(e') = ``pv11\_p1 p1a``
41. has-es-info-type(es;e';f;Id \mtimes{} pv11\_p1\_Ballot\_Num())
42. delay = 0
43. loc(e) = (fst(msgval(e')))
44. es-info-auth(es;e) = pv11\_p1\_init\_active()
45. (fst(msgval(e))) = loc(e')
46. (fst(snd(msgval(e)))) = (snd(msgval(e')))
47. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> = pv11\_p1\_AcceptorStateFun(Cmd;ldrs$\mbackslash{}ff\000C5f{uid}$;f;es;e')
\mvdash{} \mdownarrow{}l\_subset(pv11\_p1\_Ballot\_Num()
\mtimes{} \mBbbZ{}
\mtimes{} Cmd;v6;pv11\_p1\_append\_news(Cmd) pv11\_p1\_same\_pvalue(Cmd)
(snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es;pred(e))))
v6)
By
Latex:
(RepeatFor 3 (SimpEqPairs)
THEN D 0
THEN Unfold `l\_subset` 0
THEN Auto
THEN (InstLemma `pv11\_p1\_append\_news\_iff2` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}accpts\mkleeneclose{};\mkleeneopen{}ldrs\mkleeneclose{};\mkleeneopen{}ldrs$_{uid\mbackslash{}\000Cff7d$\mkleeneclose{};\mkleeneopen{}reps\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};
\mkleeneopen{}snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;bnum;es;pred(e)))\mkleeneclose{};\mkleeneopen{}v6\mkleeneclose{}]\mcdot{}
THENM (RWO "-1" 0 THEN OrRight THEN Auto)
)
THEN Auto
THEN RWO "l\_all\_iff" 0
THEN Auto
THEN GenListD (-1)
THEN DProdsAndUnions
THEN Auto)
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