Proof of Nuprl Lemma : pv11

Step * 1 1 1 2 1 of Lemma pv11_p1-agreement

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. e1 : E@i
14. e2 : E@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs_uid : Id ─→ ℤ@i
18. reps : bag(Id)@i
19. i1 : Id@i
20. i2 : Id@i
21. s : ℤ@i
22. c1 : Cmd@i
23. c2 : Cmd@i
24. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
25. Inj(Id;ℤ;ldrs_uid)@i
26. pv11_p1_decision'send(Cmd;f) i1 <s, c1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)@i
27. pv11_p1_decision'send(Cmd;f) i2 <s, c2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2)@i
28. a : Id
29. a ↓∈ accpts
30. has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
31. has-es-info-type(es;e2;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
32. ↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e2))))) (snd(snd(snd(msgval(e1))))))
33. pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e1;s;c1)
34. e' : E
35. (e' < e1)
36. loc(e') ↓∈ accpts
37. s = (fst(snd(snd(msgval(e1))))) ∈ ℤ
38. (fst(snd(msgval(e1)))) = (snd(snd(snd(msgval(e1))))) ∈ pv11_p1_Ballot_Num()
39. <loc(e1), snd(snd(snd(msgval(e1)))), s, c1> ∈ pv11_p1_p2a'base(Cmd;f)(e')
⊢ c1 = c2 ∈ Cmd
BY
{ (Assert ⌈pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e';<snd(snd(snd(msgval(e1)))), s, c1>;f)⌉⋅
   THENA (Unfold `pv11_p1_pvalue` 0
          THEN (Sel 4 (D 0) THENA Auto)
          THEN RepUR ``pv11_p1_p2a'base`` (-1)
          THEN RWO "base-classrel-equal" (-1)
          THEN Auto
          THEN D (-2)
          THEN (RevHypSubst' (-2) 0 THENA Auto)
          THEN Reduce 0
          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. e1 : E@i
14. e2 : E@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs_uid : Id ─→ ℤ@i
18. reps : bag(Id)@i
19. i1 : Id@i
20. i2 : Id@i
21. s : ℤ@i
22. c1 : Cmd@i
23. c2 : Cmd@i
24. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
25. Inj(Id;ℤ;ldrs_uid)@i
26. pv11_p1_decision'send(Cmd;f) i1 <s, c1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)@i
27. pv11_p1_decision'send(Cmd;f) i2 <s, c2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2)@i
28. a : Id
29. a ↓∈ accpts
30. has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
31. has-es-info-type(es;e2;f;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
32. ↑(pv11_p1_leq_bnum(ldrs_uid) (snd(snd(snd(msgval(e2))))) (snd(snd(snd(msgval(e1))))))
33. pv11_p1_decision{i:l}(Cmd;f;accpts;ldrs;ldrs_uid;reps;es;e1;s;c1)
34. e' : E
35. (e' < e1)
36. loc(e') ↓∈ accpts
37. s = (fst(snd(snd(msgval(e1))))) ∈ ℤ
38. (fst(snd(msgval(e1)))) = (snd(snd(snd(msgval(e1))))) ∈ pv11_p1_Ballot_Num()
39. <loc(e1), snd(snd(snd(msgval(e1)))), s, c1> ∈ pv11_p1_p2a'base(Cmd;f)(e')
40. pv11_p1_pvalue{i:l}(Cmd;ldrs_uid;accpts;es;e';<snd(snd(snd(msgval(e1)))), s, c1>;f)
⊢ c1 = c2 ∈ Cmd

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. e1 : E@i 14. e2 : E@i 15. accpts : bag(Id)@i 16. ldrs : bag(Id)@i 17. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 18. reps : bag(Id)@i 19. i1 : Id@i 20. i2 : Id@i 21. s : \mBbbZ{}@i 22. c1 : Cmd@i 23. c2 : Cmd@i 24. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 25. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 26. pv11\_p1\_decision'send(Cmd;f) i1 <s, c1> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid\mbackslash{}ff\000C7d$;reps;f)(e1)@i 27. pv11\_p1\_decision'send(Cmd;f) i2 <s, c2> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid\mbackslash{}ff\000C7d$;reps;f)(e2)@i 28. a : Id 29. a \mdownarrow{}\mmember{} accpts 30. has-es-info-type(es;e1;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num()) 31. has-es-info-type(es;e2;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num()) 32. \muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (snd(snd(snd(msgval(e2))))) (snd(snd(snd(ms\000Cgval(e1)))))) 33. pv11\_p1\_decision\{i:l\}(Cmd;f;accpts;ldrs;ldrs$_{uid}$;reps;es;e1;s;c1) 34. e' : E 35. (e' < e1) 36. loc(e') \mdownarrow{}\mmember{} accpts 37. s = (fst(snd(snd(msgval(e1))))) 38. (fst(snd(msgval(e1)))) = (snd(snd(snd(msgval(e1))))) 39. <loc(e1), snd(snd(snd(msgval(e1)))), s, c1> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e') \mvdash{} c1 = c2 By Latex: (Assert \mkleeneopen{}pv11\_p1\_pvalue\{i:l\}(Cmd;ldrs$_{uid}$;accpts;es;e';<snd(snd(snd(msgval(e\000C1)))), s, c1>f)\mkleeneclose{}\mcdot{} THENA (Unfold `pv11\_p1\_pvalue` 0 THEN (Sel 4 (D 0) THENA Auto) THEN RepUR ``pv11\_p1\_p2a'base`` (-1) THEN RWO "base-classrel-equal" (-1) THEN Auto THEN D (-2) THEN (RevHypSubst' (-2) 0 THENA Auto) THEN Reduce 0 THEN Auto) ) Home Index