Step
*
1
1
3
of Lemma
pv11_p1_unique_adopted
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. e1 : E@i
14. e2 : E@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs_uid : Id ─→ ℤ@i
18. reps : bag(Id)@i
19. bnum : pv11_p1_Ballot_Num()@i
20. pvals1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
21. pvals2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
22. proposals1 : (ℤ × Cmd) List@i
23. proposals2 : (ℤ × Cmd) List@i
24. b1 : 𝔹@i
25. b2 : 𝔹@i
26. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
27. <bnum, pvals1> ∈ pv11_p1_adopted'base(Cmd;f)(e1)@i
28. <bnum, pvals2> ∈ pv11_p1_adopted'base(Cmd;f)(e2)@i
29. <bnum, b1, proposals1> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e1)@i
30. <bnum, b2, proposals2> ∈ pv11_p1_LeaderState(Cmd;ldrs_uid;f)(e2)@i
31. (e1 <loc e2)@i
32. e' : E
33. delay : ℤ
34. (e' < e1)
35. loc(e') ↓∈ ldrs
36. (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')
37. header(e') = ``pv11_p1 p1b`` ∈ Name
38. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
39. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
40. delay = 0 ∈ ℤ
41. loc(e1) = loc(e') ∈ Id
42. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
43. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
44. msg-authentic(info(e1)) = pv11_p1_init_active()
45. bnum = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num()
46. pvals1
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
47. e3 : E
48. d1 : ℤ
49. (e3 < e2)
50. loc(e3) ↓∈ ldrs
51. e4 : {e':E| e' ≤loc e3 }
52. header(e4) = ``pv11_p1 preempted`` ∈ Name
53. has-es-info-type(es;e4;f;pv11_p1_Ballot_Num())
54. ↑(pv11_p1_is_bnum() msgval(e4))
55. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e4)) < msgval(e4))
56. (no ((pv11_p1_scout_output(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e4) loc(e4)) o pv11_p1_p1b'base(Cmd;f)) o
pv11_p1_ScoutState(Cmd;accpts;f) (pv11_p1_upd_bnum() msgval(e4) loc(e4))) between e4 and e3)
57. header(e3) = ``pv11_p1 p1b`` ∈ Name
58. has-es-info-type(es;e3;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
59. (pv11_p1_upd_bnum() msgval(e4) loc(e4)) = (fst(snd(msgval(e3)))) ∈ pv11_p1_Ballot_Num()
60. d1 = 0 ∈ ℤ
61. loc(e2) = loc(e3) ∈ Id
62. (pv11_p1_upd_bnum() msgval(e4) loc(e4)) = (fst(snd(snd(msgval(e3))))) ∈ pv11_p1_Ballot_Num()
63. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e4)
loc(e4);es.e4;e3))) < pv11_p1_threshold(accpts)
64. msg-authentic(info(e2)) = pv11_p1_init_active()
65. bnum = (pv11_p1_upd_bnum() msgval(e4) loc(e4)) ∈ pv11_p1_Ballot_Num()
66. pvals2
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e4) loc(e4);es.e4;e3)))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
⊢ pvals1 = pvals2 ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
BY
{ ((Assert ⌈False⌉⋅ THEN Auto)
THEN (HypSubst' (-22) (-2) THENA Auto)
THEN RepUR ``pv11_p1_upd_bnum pv11_p1_init_ballot_num pv11_p1_mk_bnum`` (-2)
THEN MoveToConcl (-2)
THEN GenConclAtAddrType ⌈pv11_p1_Ballot_Num()⌉ [1;3;1]⋅
THEN Auto
THEN DVar `v'
THEN DProdsAndUnions
THEN AllReduce
THEN Try (Complete (Auto))
THEN (SimpHyp (-1) THEN RepD)
THEN (InstLemma `pv11_p1_ldr_fun_pos` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈e4⌉;⌈ldrs_uid⌉]⋅ THENA Auto)
THEN (FLemma `pv11_p1_lt_bnum_trans2` [-1;-17] THENA Auto)
THEN (HypSubst (-5) (-1) THENA Auto)
THEN Repeat (SimpleAssertReasoning)
THEN RW assert_pushdownC (-1)
THEN DProdsAndUnions⋅
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. e1 : E@i
14. e2 : E@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
18. reps : bag(Id)@i
19. bnum : pv11\_p1\_Ballot\_Num()@i
20. pvals1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
21. pvals2 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i
22. proposals1 : (\mBbbZ{} \mtimes{} Cmd) List@i
23. proposals2 : (\mBbbZ{} \mtimes{} Cmd) List@i
24. b1 : \mBbbB{}@i
25. b2 : \mBbbB{}@i
26. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
27. <bnum, pvals1> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e1)@i
28. <bnum, pvals2> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e2)@i
29. <bnum, b1, proposals1> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e1)@i
30. <bnum, b2, proposals2> \mmember{} pv11\_p1\_LeaderState(Cmd;ldrs$_{uid}$;f)(e2)@i
31. (e1 <loc e2)@i
32. e' : E
33. delay : \mBbbZ{}
34. (e' < e1)
35. loc(e') \mdownarrow{}\mmember{} ldrs
36. (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')
37. header(e') = ``pv11\_p1 p1b``
38. 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))
39. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(msgval(e'))))
40. delay = 0
41. loc(e1) = loc(e')
42. (pv11\_p1\_init\_ballot\_num() loc(e')) = (fst(snd(snd(msgval(e')))))
43. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num()
loc(e');es;e'))) < pv11\_p1\_threshold(accpts)
44. msg-authentic(info(e1)) = pv11\_p1\_init\_active()
45. bnum = (pv11\_p1\_init\_ballot\_num() loc(e'))
46. pvals1 = (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_init\_ballot\_num() loc(e');es;e')))
47. e3 : E
48. d1 : \mBbbZ{}
49. (e3 < e2)
50. loc(e3) \mdownarrow{}\mmember{} ldrs
51. e4 : \{e':E| e' \mleq{}loc e3 \}
52. header(e4) = ``pv11\_p1 preempted``
53. has-es-info-type(es;e4;f;pv11\_p1\_Ballot\_Num())
54. \muparrow{}(pv11\_p1\_is\_bnum() msgval(e4))
55. \muparrow{}(fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e4)) < msgval(e4))
56. (no ((pv11\_p1\_scout\_output(Cmd;accpts;f) (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4)) o
pv11\_p1\_p1b'base(Cmd;f)) o pv11\_p1\_ScoutState(Cmd;accpts;f)
(pv11\_p1\_upd\_bnum() msgval(e4) loc(e4))) between e4 and e3)
57. header(e3) = ``pv11\_p1 p1b``
58. has-es-info-type(es;e3;f;Id
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} pv11\_p1\_Ballot\_Num()
\mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
59. (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4)) = (fst(snd(msgval(e3))))
60. d1 = 0
61. loc(e2) = loc(e3)
62. (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4)) = (fst(snd(snd(msgval(e3)))))
63. \#(fst(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e4)
loc(e4);es.e4;e3))) < pv11\_p1\_threshold(accpts)
64. msg-authentic(info(e2)) = pv11\_p1\_init\_active()
65. bnum = (pv11\_p1\_upd\_bnum() msgval(e4) loc(e4))
66. pvals2
= (snd(pv11\_p1\_ScoutStateFun(Cmd;accpts;f;pv11\_p1\_upd\_bnum() msgval(e4) loc(e4);es.e4;e3)))
\mvdash{} pvals1 = pvals2
By
Latex:
((Assert \mkleeneopen{}False\mkleeneclose{}\mcdot{} THEN Auto)
THEN (HypSubst' (-22) (-2) THENA Auto)
THEN RepUR ``pv11\_p1\_upd\_bnum pv11\_p1\_init\_ballot\_num pv11\_p1\_mk\_bnum`` (-2)
THEN MoveToConcl (-2)
THEN GenConclAtAddrType \mkleeneopen{}pv11\_p1\_Ballot\_Num()\mkleeneclose{} [1;3;1]\mcdot{}
THEN Auto
THEN DVar `v'
THEN DProdsAndUnions
THEN AllReduce
THEN Try (Complete (Auto))
THEN (SimpHyp (-1) THEN RepD)
THEN (InstLemma `pv11\_p1\_ldr\_fun\_pos` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e4\mkleeneclose{};\mkleeneopen{}ldrs$_{uid}$\mkleeneclose{}]\mcdot{} THE\000CNA Auto)
THEN (FLemma `pv11\_p1\_lt\_bnum\_trans2` [-1;-17] THENA Auto)
THEN (HypSubst (-5) (-1) THENA Auto)
THEN Repeat (SimpleAssertReasoning)
THEN RW assert\_pushdownC (-1)
THEN DProdsAndUnions\mcdot{}
THEN Auto)
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