Step
*
1
2
2
2
of Lemma
pv11_p1_bnum_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. ldrs_uid : Id ─→ ℤ@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. reps : bag(Id)@i
18. b : pv11_p1_Ballot_Num()@i
19. i : Id@i
20. l : Id@i
21. s : ℤ@i
22. c : Cmd@i
23. loc(e) ↓∈ ldrs
24. i ↓∈ accpts
25. 0 = 0 ∈ ℤ
26. ff = pv11_p1_init_active()
27. l = loc(e) ∈ Id
28. header(e) = ``pv11_p1 adopted`` ∈ Name
29. has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
30. (fst(msgval(e))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))) ∈ pv11_p1_Ballot_Num()
31. <s, c> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))
32. ¬(∃p2:Cmd. (<s, p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e)))))
33. b = (fst(msgval(e))) ∈ pv11_p1_Ballot_Num()
34. (<s, c> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))))
⊢ ∃n:ℤ. (↑(pv11_p1_eq_bnums() b (pv11_p1_mk_bnum() n l)))
BY
{ ((InstLemma `pv11_p1_ldr_fun_loc_bnum` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈e⌉;⌈ldrs_uid⌉]⋅ THENA Auto) THEN ParallelLast 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. e : E@i
14. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. reps : bag(Id)@i
18. b : pv11\_p1\_Ballot\_Num()@i
19. i : Id@i
20. l : Id@i
21. s : \mBbbZ{}@i
22. c : Cmd@i
23. loc(e) \mdownarrow{}\mmember{} ldrs
24. i \mdownarrow{}\mmember{} accpts
25. 0 = 0
26. ff = pv11\_p1\_init\_active()
27. l = loc(e)
28. header(e) = ``pv11\_p1 adopted``
29. has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))
30. (fst(msgval(e))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))
31. <s, c> \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))
32. \mneg{}(\mexists{}p2:Cmd. (<s, p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) (snd(msgval(e)))))
33. b = (fst(msgval(e)))
34. (<s, c> \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))))
\mvdash{} \mexists{}n:\mBbbZ{}. (\muparrow{}(pv11\_p1\_eq\_bnums() b (pv11\_p1\_mk\_bnum() n l)))
By
Latex:
((InstLemma `pv11\_p1\_ldr\_fun\_loc\_bnum` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}ldrs$_{uid}$\mkleeneclose{}]\mcdot{} THEN\000CA Auto)
THEN ParallelLast
THEN Auto)
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