Step
*
1
2
2
1
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. b : pv11_p1_Ballot_Num()@i'
24. l : Id
25. r : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
26. header(e) = ``pv11_p1 p1b`` ∈ Name
27. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
28. <l, b, b, r> = msgval(e) ∈ (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) Lis\000Ct))
29. (<p1, p3, p4> ∈ r)
⊢ ↓∃e':E. (e' c≤ e ∧ pv11_p1_from-p2a{i:l}(Cmd;es;e';<p1, p3, p4>))
BY
{ (UseMessageConstraint (-4) THEN SqExRepD) }
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. b : pv11_p1_Ballot_Num()@i'
24. l : Id
25. r : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
26. header(e) = ``pv11_p1 p1b`` ∈ Name
27. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
28. <l, b, b, r> = msgval(e) ∈ (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) Lis\000Ct))
29. (<p1, p3, p4> ∈ r)
30. ∀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'))))
31. e' : E
32. delay : ℤ
33. (e' < e)
34. loc(e') ↓∈ accpts
35. header(e') = ``pv11_p1 p1a`` ∈ Name
36. has-es-info-type(es;e';f;Id × pv11_p1_Ballot_Num())
37. delay = 0 ∈ ℤ
38. loc(e) = (fst(msgval(e'))) ∈ Id
39. es-info-auth(es;e) = pv11_p1_init_active()
40. (fst(msgval(e))) = loc(e') ∈ Id
41. (fst(snd(msgval(e)))) = (snd(msgval(e'))) ∈ pv11_p1_Ballot_Num()
42. <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))
⊢ ↓∃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. b : pv11\_p1\_Ballot\_Num()@i'
24. l : Id
25. r : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List
26. header(e) = ``pv11\_p1 p1b``
27. 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))
28. <l, b, b, r> = msgval(e)
29. (<p1, p3, p4> \mmember{} r)
\mvdash{} \mdownarrow{}\mexists{}e':E. (e' c\mleq{} e \mwedge{} pv11\_p1\_from-p2a\{i:l\}(Cmd;es;e';<p1, p3, p4>))
By
Latex:
(UseMessageConstraint (-4) THEN SqExRepD)
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