Proof of Nuprl Lemma : pv11

Step * 1 1 of Lemma pv11_p1-agreement2

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. Inj(Id;ℤ;ldrs_uid)@i
19. e1 : E@i
20. e2 : E@i
21. v1 : ℤ × Cmd@i
22. v2 : ℤ × Cmd@i
23. v1 ∈ pv11_p1_decision'base(Cmd;f)(e1)@i
24. v2 ∈ pv11_p1_decision'base(Cmd;f)(e2)@i
25. (fst(v1)) = (fst(v2)) ∈ ℤ@i
26. e' : E
27. delay : ℤ
28. (e' < e1)
29. make-msg-interface(delay;loc(e1);mk-msg(msg-authentic(info(e1));[decision];v1)) ∈
     pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e')
30. e3 : E
31. d1 : ℤ
32. (e3 < e2)
33. make-msg-interface(d1;loc(e2);mk-msg(msg-authentic(info(e2));[decision];v2)) ∈
     pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e3)
⊢ (snd(v1)) = (snd(v2)) ∈ Cmd
BY
{ (DProds
   THEN AllReduce
   THEN (Subst ⌈delay ~ 0⌉ (-5)⋅
         THENA (Try ((RWO "make-Msg-as-mk-msg" (-5) THENA Auto))
                THEN RewriteWithILF (-5)
                THEN Auto
                THEN ExRepD
                THEN Auto)
         )
   THEN (Subst ⌈d1 ~ 0⌉ (-1)⋅
         THENA (Try ((RWO "make-Msg-as-mk-msg" (-1) THENA Auto))
                THEN RewriteWithILF (-1)
                THEN Auto
                THEN ExRepD
                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

31. make-msg-interface(0;loc(e1);mk-msg(msg-authentic(info(e1));[decision];<v5, v6>)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldr\000Cs_uid;reps;f)(e')

32. e3 : E
33. d1 : ℤ
34. (e3 < e2)

35. make-msg-interface(0;loc(e2);mk-msg(msg-authentic(info(e2));[decision];<v3, v4>)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldr\000Cs_uid;reps;f)(e3)

⊢ v6 = v4 ∈ 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. 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. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 19. e1 : E@i 20. e2 : E@i 21. v1 : \mBbbZ{} \mtimes{} Cmd@i 22. v2 : \mBbbZ{} \mtimes{} Cmd@i 23. v1 \mmember{} pv11\_p1\_decision'base(Cmd;f)(e1)@i 24. v2 \mmember{} pv11\_p1\_decision'base(Cmd;f)(e2)@i 25. (fst(v1)) = (fst(v2))@i 26. e' : E 27. delay : \mBbbZ{} 28. (e' < e1) 29. make-msg-interface(delay;loc(e1);mk-msg(msg-authentic(info(e1));[decision];v1)) \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}$;reps;f)(e') 30. e3 : E 31. d1 : \mBbbZ{} 32. (e3 < e2) 33. make-msg-interface(d1;loc(e2);mk-msg(msg-authentic(info(e2));[decision];v2)) \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}$;reps;f)(e3) \mvdash{} (snd(v1)) = (snd(v2)) By Latex: (DProds THEN AllReduce THEN (Subst \mkleeneopen{}delay \msim{} 0\mkleeneclose{} (-5)\mcdot{} THENA (Try ((RWO "make-Msg-as-mk-msg" (-5) THENA Auto)) THEN RewriteWithILF (-5) THEN Auto THEN ExRepD THEN Auto) ) THEN (Subst \mkleeneopen{}d1 \msim{} 0\mkleeneclose{} (-1)\mcdot{} THENA (Try ((RWO "make-Msg-as-mk-msg" (-1) THENA Auto)) THEN RewriteWithILF (-1) THEN Auto THEN ExRepD THEN Auto) )) Home Index