Proof of Nuprl Lemma : pv11_p1_acc_rcv

Step * 1 4 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

27. ∃e':E. ∃i,l:Id. pv11_p1_p2a'send(Cmd;f) i <l, snd(msgval(e))> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e')@i

28. ↑e ∈_b pv11_p1_p2a'base(Cmd;f)
29. ↑e ∈_b pv11_p1_p2a'base(Cmd;f)
30. ↑(pv11_p1_leq_bnum(ldrs_uid) (fst(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) (fst(snd(msgval(e)))))
31. (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
⊢ ↓(<fst(snd(msgval(e))), snd(snd(msgval(e)))> ∈ pv11_p1_add_if_new() pv11_p1_same_pvalue(Cmd)

                                                 <fst(snd(msgval(e))), snd(snd(msgval(e)))>

                                                 (snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))))

BY
{ ((RWO "pv11_p1_add_if_new_iff2" 0 THENA Auto)
   THEN AutoSplit
   THEN (RWO "l_exists_iff" (-1) THENA Auto)
   THEN ExRepD
   THEN (FLemma `pv11_p1_acc_fun_p2a_pv` [-2] THENA Auto)
   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. 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. z : pv11_p1_Ballot_Num() × ℤ × Cmd
36. (z ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))))
37. ↑(pv11_p1_same_pvalue(Cmd) <fst(snd(msgval(e))), snd(snd(msgval(e)))> z)
38. e1 : E
39. l1 : Id
40. e1 ≤loc pred(e)
41. <l1, z> ∈ pv11_p1_p2a'base(Cmd;f)(e1)

⊢ ↓(<fst(snd(msgval(e))), snd(snd(msgval(e)))> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e)))) ∨ False

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. \mexists{}e':E. \mexists{}i,l:Id. pv11\_p1\_p2a'send(Cmd;f) i <l, snd(msgval(e))> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldr\000Cs$_{uid}$;reps;f)(e')@i 28. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f) 29. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f) 30. \muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$\000C_{uid}$;f;es;pred(e)))) (fst(snd(msgval(e))))) 31. (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 \mvdash{} \mdownarrow{}(<fst(snd(msgval(e))), snd(snd(msgval(e)))> \mmember{} pv11\_p1\_add\_if\_new() pv11\_p1\_same\_pvalue(Cmd) <fst(snd(msgval(e))), snd(snd(msgval(e)))> (snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e))))) By Latex: ((RWO "pv11\_p1\_add\_if\_new\_iff2" 0 THENA Auto) THEN AutoSplit THEN (RWO "l\_exists\_iff" (-1) THENA Auto) THEN ExRepD THEN (FLemma `pv11\_p1\_acc\_fun\_p2a\_pv` [-2] THENA Auto) THEN ExRepD) Home Index