Step
*
1
2
1
1
2
1
of Lemma
pv11_p1_A5_C2
.....antecedent.....
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. b : pv11_p1_Ballot_Num()@i
18. s : ℤ@i
19. p : Cmd@i
20. p' : Cmd@i
21. as : bag(Id)@i
22. cs : bag(Id)@i
23. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
24. Inj(Id;ℤ;ldrs_uid)@i
25. accpts = (as + cs) ∈ bag(Id)@i
26. #(as) < pv11_p1_threshold(accpts)@i
27. ∀a:Id
(a ↓∈ cs ⇒ (↓∃e':E. ((a = loc(e') ∈ Id) ∧ (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e'))))))@i
28. e : E@i
29. ∀e1:E
((e1 < e)
⇒ (∀b':pv11_p1_Ballot_Num()
((↑(b < b'))
⇒ (∀l,i:Id.
(((<b', s, p'> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e1)))
∨ pv11_p1_p2a'send(Cmd;f) i <l, b', s, p'> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1))
⇒ (p = p' ∈ Cmd))))))
30. b' : pv11_p1_Ballot_Num()@i
31. ↑(b < b')@i
32. l : Id@i
33. i : Id@i
34. loc(e) ↓∈ ldrs
35. i ↓∈ accpts
36. 0 = 0 ∈ ℤ
37. ff = pv11_p1_init_active()
38. l = loc(e) ∈ Id
39. header(e) = [propose] ∈ Name
40. has-es-info-type(es;e;f;ℤ × Cmd)
41. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))))
42. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))))
43. b' = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))) ∈ pv11_p1_Ballot_Num()
44. <s, p'> = msgval(e) ∈ (ℤ × Cmd)
45. e' : E
46. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
47. (e' <loc e)
48. <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)), pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e')
49. (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))
= (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))
∈ pv11_p1_Ballot_Num()
50. L : E List
51. #([i∈accpts|¬bi ∈b map(λe.loc(e);L))]) < pv11_p1_threshold(accpts)
52. ∀pv:pv11_p1_Ballot_Num() × ℤ × Cmd
((pv ∈ pvals) ⇐⇒ (pv ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L))))
53. ∀e1:E
((e1 ∈ L)
⇒ ((e1 < e')
∧ loc(e1) ↓∈ accpts
∧ ((fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))
= (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e1)))
∈ pv11_p1_Ballot_Num())))
54. i1 : Id
55. v : E
56. v ↓∈ L
57. i1 = loc(v) ∈ Id
58. i1 ↓∈ cs
59. e1 : E
60. i1 = loc(e1) ∈ Id
61. (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e1)))
62. (v < e')
63. loc(v) ↓∈ accpts
64. (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))
= (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v)))
∈ pv11_p1_Ballot_Num()
65. b' = (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v))) ∈ pv11_p1_Ballot_Num()
66. (<b, s, p> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v)))
⊢ (<b, s, p> ∈ concat(map(λe.(snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)));L)))
BY
{ (GenListD 0
THEN InstConcl [⌈snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;v))⌉]⋅
THEN Auto
THEN GenListD 0
THEN InstConcl [⌈v⌉]⋅
THEN Auto
THEN (FLemma `bag-member-sq-list-member` [-11] THENA Auto)
THEN TrySquashExRepD (-1)
THEN Auto) }
Latex:
Latex:
.....antecedent.....
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. b : pv11\_p1\_Ballot\_Num()@i
18. s : \mBbbZ{}@i
19. p : Cmd@i
20. p' : Cmd@i
21. as : bag(Id)@i
22. cs : bag(Id)@i
23. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
24. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
25. accpts = (as + cs)@i
26. \#(as) < pv11\_p1\_threshold(accpts)@i
27. \mforall{}a:Id
(a \mdownarrow{}\mmember{} cs
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
((a = loc(e')) \mwedge{} (<b, s, p> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid\mbackslash{}ff\000C7d$;f;es;e'))))))@i
28. e : E@i
29. \mforall{}e1:E
((e1 < e)
{}\mRightarrow{} (\mforall{}b':pv11\_p1\_Ballot\_Num()
((\muparrow{}(b < b'))
{}\mRightarrow{} (\mforall{}l,i:Id.
(((<b', s, p'> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$\000C;f;es;e1)))
\mvee{} pv11\_p1\_p2a'send(Cmd;f) i <l, b', s, p'> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs\mbackslash{}ff2\000C4_{uid}$;reps;f)(e1))
{}\mRightarrow{} (p = p'))))))
30. b' : pv11\_p1\_Ballot\_Num()@i
31. \muparrow{}(b < b')@i
32. l : Id@i
33. i : Id@i
34. loc(e) \mdownarrow{}\mmember{} ldrs
35. i \mdownarrow{}\mmember{} accpts
36. 0 = 0
37. ff = pv11\_p1\_init\_active()
38. l = loc(e)
39. header(e) = [propose]
40. has-es-info-type(es;e;f;\mBbbZ{} \mtimes{} Cmd)
41. \muparrow{}(fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))))
42. \mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) (fst(msgval(e)))
(snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))))
43. b' = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))
44. <s, p'> = msgval(e)
45. e' : E
46. pvals : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
47. (e' <loc e)
48. <fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)), pvals> \mmember{} pv11\_p1\_adop\000Cted'base(Cmd;f)(e')
49. (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))
= (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e')))
50. L : E List
51. \#([i\mmember{}accpts|\mneg{}\msubb{}i \mmember{}\msubb{} map(\mlambda{}e.loc(e);L))]) < pv11\_p1\_threshold(accpts)
52. \mforall{}pv:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd
((pv \mmember{} pvals)
\mLeftarrow{}{}\mRightarrow{} (pv \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;e\000Cs;e)));L))))
53. \mforall{}e1:E
((e1 \mmember{} L)
{}\mRightarrow{} ((e1 < e')
\mwedge{} loc(e1) \mdownarrow{}\mmember{} accpts
\mwedge{} ((fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))
= (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e1))))))
54. i1 : Id
55. v : E
56. v \mdownarrow{}\mmember{} L
57. i1 = loc(v)
58. i1 \mdownarrow{}\mmember{} cs
59. e1 : E
60. i1 = loc(e1)
61. (<b, s, p> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e1)))
62. (v < e')
63. loc(v) \mdownarrow{}\mmember{} accpts
64. (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))
= (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;v)))
65. b' = (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;v)))
66. (<b, s, p> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;v)))
\mvdash{} (<b, s, p> \mmember{} concat(map(\mlambda{}e.(snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es\000C;e)));L)))
By
Latex:
(GenListD 0
THEN InstConcl [\mkleeneopen{}snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;v))\mkleeneclose{}]\mcdot{}
THEN Auto
THEN GenListD 0
THEN InstConcl [\mkleeneopen{}v\mkleeneclose{}]\mcdot{}
THEN Auto
THEN (FLemma `bag-member-sq-list-member` [-11] THENA Auto)
THEN TrySquashExRepD (-1)
THEN Auto)
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