Step
*
1
4
1
1
1
of Lemma
pv11_p1_acc_rcv_p2a
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. ¬↑e ∈b pv11_p1_p1a'base(Cmd;f)
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ─→ ℤ@i
19. reps : bag(Id)@i
20. b : pv11_p1_Ballot_Num()@i
21. s : ℤ@i
22. c : Cmd@i
23. Inj(Id;ℤ;ldrs_uid)@i
24. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
25. header(e) = ``pv11_p1 p2a`` ∈ Name@i
26. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i
27. e' : E@i
28. i : Id@i
29. l : Id@i
30. pv11_p1_p2a'send(Cmd;f) i <l, snd(msgval(e))> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e')@i
31. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
32. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
33. ↑(pv11_p1_leq_bnum(ldrs_uid) (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) (fst(snd(msgval(e)))))
34. (pv11_p1_max_bnum(ldrs_uid) (fst(snd(msgval(e)))) (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))))
= (fst(snd(msgval(e))))
∈ pv11_p1_Ballot_Num()@i
35. z1 : pv11_p1_Ballot_Num()
36. z3 : ℤ
37. z4 : Cmd
38. (<z1, z3, z4> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))))
39. ↑(pv11_p1_same_pvalue(Cmd) <fst(snd(msgval(e))), snd(snd(msgval(e)))> <z1, z3, z4>)
40. e1 : E
41. l1 : Id
42. e1 ≤loc pred(e)
43. <l1, z1, z3, z4> ∈ pv11_p1_p2a'base(Cmd;f)(e1)
44. header(e1) = ``pv11_p1 p2a`` ∈ Name
45. has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
46. <l1, z1, z3, z4> = msgval(e1) ∈ (Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
47. ∀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'))))
48. e2 : E
49. delay : ℤ
50. (e2 < e1)
51. pv11_p1_p2a'send(Cmd;f) loc(e1) msgval(e1) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2)@i
52. msgval(e) ∈ Id × pv11_p1_Ballot_Num() × ℤ × Cmd
⊢ (<b, []> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e')
∧ <b, []> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e2)
∧ (<z1, z3, snd(snd(snd(msgval(e))))> ∈ [])
∧ (<z1, z3, z4> ∈ []))
∨ (pv11_p1_p2a'send(Cmd;f) i <l1, z1, z3, snd(snd(snd(msgval(e))))> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e')
∧ pv11_p1_p2a'send(Cmd;f) loc(e1) <l1, z1, z3, z4> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2))
BY
{ (OrRight 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. ¬↑e ∈b pv11_p1_p1a'base(Cmd;f)
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs_uid : Id ─→ ℤ@i
19. reps : bag(Id)@i
20. b : pv11_p1_Ballot_Num()@i
21. s : ℤ@i
22. c : Cmd@i
23. Inj(Id;ℤ;ldrs_uid)@i
24. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
25. header(e) = ``pv11_p1 p2a`` ∈ Name@i
26. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i
27. e' : E@i
28. i : Id@i
29. l : Id@i
30. pv11_p1_p2a'send(Cmd;f) i <l, snd(msgval(e))> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e')@i
31. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
32. ↑e ∈b pv11_p1_p2a'base(Cmd;f)
33. ↑(pv11_p1_leq_bnum(ldrs_uid) (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) (fst(snd(msgval(e)))))
34. (pv11_p1_max_bnum(ldrs_uid) (fst(snd(msgval(e)))) (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))))
= (fst(snd(msgval(e))))
∈ pv11_p1_Ballot_Num()@i
35. z1 : pv11_p1_Ballot_Num()
36. z3 : ℤ
37. z4 : Cmd
38. (<z1, z3, z4> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))))
39. ↑(pv11_p1_same_pvalue(Cmd) <fst(snd(msgval(e))), snd(snd(msgval(e)))> <z1, z3, z4>)
40. e1 : E
41. l1 : Id
42. e1 ≤loc pred(e)
43. <l1, z1, z3, z4> ∈ pv11_p1_p2a'base(Cmd;f)(e1)
44. header(e1) = ``pv11_p1 p2a`` ∈ Name
45. has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
46. <l1, z1, z3, z4> = msgval(e1) ∈ (Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
47. ∀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'))))
48. e2 : E
49. delay : ℤ
50. (e2 < e1)
51. pv11_p1_p2a'send(Cmd;f) loc(e1) msgval(e1) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2)@i
52. msgval(e) ∈ Id × pv11_p1_Ballot_Num() × ℤ × Cmd
⊢ pv11_p1_p2a'send(Cmd;f) i <l1, z1, z3, snd(snd(snd(msgval(e))))> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(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. \mneg{}\muparrow{}e \mmember{}\msubb{} pv11\_p1\_p1a'base(Cmd;f)
16. accpts : bag(Id)@i
17. ldrs : bag(Id)@i
18. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
19. reps : bag(Id)@i
20. b : pv11\_p1\_Ballot\_Num()@i
21. s : \mBbbZ{}@i
22. c : Cmd@i
23. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i
24. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
25. header(e) = ``pv11\_p1 p2a``@i
26. has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)@i
27. e' : E@i
28. i : Id@i
29. l : Id@i
30. pv11\_p1\_p2a'send(Cmd;f) i <l, snd(msgval(e))> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{\000Cuid}$;reps;f)(e')@i
31. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f)
32. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f)
33. \muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$\000C_{uid}$;f;es;pred(e))))
(fst(snd(msgval(e)))))
34. (pv11\_p1\_max\_bnum(ldrs$_{uid}$) (fst(snd(msgval(e))))
(fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e)))))
= (fst(snd(msgval(e))))@i
35. z1 : pv11\_p1\_Ballot\_Num()
36. z3 : \mBbbZ{}
37. z4 : Cmd
38. (<z1, z3, z4> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e)))\000C)
39. \muparrow{}(pv11\_p1\_same\_pvalue(Cmd) <fst(snd(msgval(e))), snd(snd(msgval(e)))> <z1, z3, z4>)
40. e1 : E
41. l1 : Id
42. e1 \mleq{}loc pred(e)
43. <l1, z1, z3, z4> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e1)
44. header(e1) = ``pv11\_p1 p2a``
45. has-es-info-type(es;e1;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd)
46. <l1, z1, z3, z4> = msgval(e1)
47. \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'))))
48. e2 : E
49. delay : \mBbbZ{}
50. (e2 < e1)
51. pv11\_p1\_p2a'send(Cmd;f) loc(e1) msgval(e1) \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid\000C}$;reps;f)(e2)@i
52. msgval(e) \mmember{} Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd
\mvdash{} (<b, []> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e')
\mwedge{} <b, []> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e2)
\mwedge{} (<z1, z3, snd(snd(snd(msgval(e))))> \mmember{} [])
\mwedge{} (<z1, z3, z4> \mmember{} []))
\mvee{} (pv11\_p1\_p2a'send(Cmd;f) i <l1, z1, z3, snd(snd(snd(msgval(e))))> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;l\000Cdrs$_{uid}$;reps;f)(e')
\mwedge{} pv11\_p1\_p2a'send(Cmd;f) loc(e1) <l1, z1, z3, z4> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_\mbackslash{}f\000Cf7buid}$;reps;f)(e2))
By
Latex:
(OrRight THEN Auto)
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