Step
*
1
4
of Lemma
pv11_p1_pvalue_from_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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
((e1 < e)
⇒ (∀p:pv11_p1_Ballot_Num() × ℤ × Cmd
(pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e1;p;f)
⇒ (↓∃e':E. (e' c≤ e1 ∧ pv11_p1_from-p2a{i:l}(Cmd;es;e';p))))))
20. p1 : pv11_p1_Ballot_Num()@i
21. p3 : ℤ@i
22. p4 : Cmd@i
23. header(e) = ``pv11_p1 adopted`` ∈ Name@i'
24. (<p1, p3, p4> ∈ snd(msgval(e)))@i'
⊢ ↓∃e':E. (e' c≤ e ∧ pv11_p1_from-p2a{i:l}(Cmd;es;e';<p1, p3, p4>))
BY
{ ((Assert ⌈has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))⌉⋅ THENA Auto)
THEN UseMessageConstraint (-3)
THEN Try (Complete (Repeat ((RWO "pair-eta<" 0 THEN Auto))))
THEN SqExRepD
THEN D (-1)
THEN ExRepD) }
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
((e1 < e)
⇒ (∀p:pv11_p1_Ballot_Num() × ℤ × Cmd
(pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e1;p;f)
⇒ (↓∃e':E. (e' c≤ e1 ∧ pv11_p1_from-p2a{i:l}(Cmd;es;e';p))))))
20. p1 : pv11_p1_Ballot_Num()@i
21. p3 : ℤ@i
22. p4 : Cmd@i
23. header(e) = ``pv11_p1 adopted`` ∈ Name@i'
24. (<p1, p3, p4> ∈ snd(msgval(e)))@i'
25. has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
26. ∀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'))))
27. e' : E
28. delay : ℤ
29. (e' < e)
30. loc(e') ↓∈ ldrs
31. (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')
32. header(e') = ``pv11_p1 p1b`` ∈ Name
33. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
34. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
35. delay = 0 ∈ ℤ
36. loc(e) = loc(e') ∈ Id
37. (pv11_p1_init_ballot_num() loc(e')) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
38. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e'))) < pv11_p1_threshold(accpts)
39. es-info-auth(es;e) = pv11_p1_init_active()
40. (fst(msgval(e))) = (pv11_p1_init_ballot_num() loc(e')) ∈ pv11_p1_Ballot_Num()
41. (snd(msgval(e)))
= (snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_init_ballot_num() loc(e');es;e')))
∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
⊢ ↓∃e':E. (e' c≤ e ∧ pv11_p1_from-p2a{i:l}(Cmd;es;e';<p1, p3, p4>))
2
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. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
((e1 < e)
⇒ (∀p:pv11_p1_Ballot_Num() × ℤ × Cmd
(pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e1;p;f)
⇒ (↓∃e':E. (e' c≤ e1 ∧ pv11_p1_from-p2a{i:l}(Cmd;es;e';p))))))
20. p1 : pv11_p1_Ballot_Num()@i
21. p3 : ℤ@i
22. p4 : Cmd@i
23. header(e) = ``pv11_p1 adopted`` ∈ Name@i'
24. (<p1, p3, p4> ∈ snd(msgval(e)))@i'
25. has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
26. ∀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'))))
27. e' : E
28. delay : ℤ
29. (e' < e)
30. loc(e') ↓∈ ldrs
31. e1 : {e1:E| e1 ≤loc e' }
32. header(e1) = ``pv11_p1 preempted`` ∈ Name
33. has-es-info-type(es;e1;f;pv11_p1_Ballot_Num())
34. ↑(pv11_p1_is_bnum() msgval(e1))
35. ↑(fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)) < msgval(e1))
36. (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')
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_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(msgval(e')))) ∈ pv11_p1_Ballot_Num()
40. delay = 0 ∈ ℤ
41. loc(e) = loc(e') ∈ Id
42. (pv11_p1_upd_bnum() msgval(e1) loc(e1)) = (fst(snd(snd(msgval(e'))))) ∈ pv11_p1_Ballot_Num()
43. #(fst(pv11_p1_ScoutStateFun(Cmd;accpts;f;pv11_p1_upd_bnum() msgval(e1)
loc(e1);es.e1;e'))) < pv11_p1_threshold(accpts)
44. es-info-auth(es;e) = pv11_p1_init_active()
45. (fst(msgval(e))) = (pv11_p1_upd_bnum() msgval(e1) loc(e1)) ∈ pv11_p1_Ballot_Num()
46. (snd(msgval(e)))
= (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. (e' c≤ e ∧ pv11_p1_from-p2a{i:l}(Cmd;es;e';<p1, p3, p4>))
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
16. reps : bag(Id)@i
17. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i
18. e : E@i
19. \mforall{}e1:E
((e1 < e)
{}\mRightarrow{} (\mforall{}p:pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd
(pv11\_p1\_pvalue\{i:l\}(Cmd;ldrs$_{uid}$;accpts;es;e1;p;f)
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E. (e' c\mleq{} e1 \mwedge{} pv11\_p1\_from-p2a\{i:l\}(Cmd;es;e';p))))))
20. p1 : pv11\_p1\_Ballot\_Num()@i
21. p3 : \mBbbZ{}@i
22. p4 : Cmd@i
23. header(e) = ``pv11\_p1 adopted``@i'
24. (<p1, p3, p4> \mmember{} snd(msgval(e)))@i'
\mvdash{} \mdownarrow{}\mexists{}e':E. (e' c\mleq{} e \mwedge{} pv11\_p1\_from-p2a\{i:l\}(Cmd;es;e';<p1, p3, p4>))
By
Latex:
((Assert \mkleeneopen{}has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))\mkleeneclose{}\mcdot{}
THENA Auto
)
THEN UseMessageConstraint (-3)
THEN Try (Complete (Repeat ((RWO "pair-eta<" 0 THEN Auto))))
THEN SqExRepD
THEN D (-1)
THEN ExRepD)
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