Proof of Nuprl Lemma : pv11_p1_A4

Step * 1 1 1 of Lemma pv11_p1_A4_C1

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. Inj(Id;ℤ;ldrs_uid)@i
18. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
19. e2 : E@i
20. ∀e3:E
((e3 < e2)
⇒

 (∀e1:E. ∀bnum1,bnum2:pv11_p1_Ballot_Num(). ∀accepted1,accepted2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.

          ∀i1,i2,l:Id. ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀p1,p2:Cmd.
            (((<bnum1, accepted1> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e1)
            ∧ <bnum2, accepted2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e3)
            ∧ (<b, s, p1> ∈ accepted1)
            ∧ (<b, s, p2> ∈ accepted2))

            ∨ (pv11_p1_p2a'send(Cmd;f) i1 <l, b, s, p1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)

              ∧ pv11_p1_p2a'send(Cmd;f) i2 <l, b, s, p2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e3)))

⇒ (p1 = p2 ∈ Cmd))))
21. e1 : E@i
22. bnum1 : pv11_p1_Ballot_Num()@i
23. bnum2 : pv11_p1_Ballot_Num()@i
24. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
25. accepted2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. i1 : Id@i
27. i2 : Id@i
28. l : Id@i
29. b : pv11_p1_Ballot_Num()@i
30. s : ℤ@i
31. p1 : Cmd@i
32. p2 : Cmd@i
33. <bnum1, accepted1> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e1)@i
34. <bnum2, accepted2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e2)@i
35. (<b, s, p1> ∈ accepted1)@i
36. (<b, s, p2> ∈ accepted2)@i
37. e3 : E
38. l1 : Id
39. e3 ≤loc e1
40. <l1, b, s, p1> ∈ pv11_p1_p2a'base(Cmd;f)(e3)
41. e4 : E
42. l2 : Id
43. e4 ≤loc e2
44. <l2, b, s, p2> ∈ pv11_p1_p2a'base(Cmd;f)(e4)
45. e' : E
46. delay : ℤ
47. (e' < e3)

48. make-msg-interface(delay;loc(e3);mk-msg(msg-authentic(info(e3));``pv11_p1 p2a``;<l1, b, s, p1>)) ∈ pv11_p1_main(Cmd;\000Caccpts;ldrs;ldrs_uid;reps;f)(e')

49. e5 : E
50. d1 : ℤ
51. (e5 < e4)

52. make-msg-interface(d1;loc(e4);mk-msg(msg-authentic(info(e4));``pv11_p1 p2a``;<l2, b, s, p2>)) ∈ pv11_p1_main(Cmd;acc\000Cpts;ldrs;ldrs_uid;reps;f)(e5)

⊢ p1 = p2 ∈ Cmd
BY
{ ((Subst ⌈delay ~ 0⌉ (-5)⋅ THENA (RewriteWithILF (-5) THEN Auto))
   THEN (Subst ⌈d1 ~ 0⌉ (-1)⋅ THENA (RewriteWithILF (-1) THEN Auto))
   THEN (Subst ⌈msg-authentic(info(e3)) ~ ff⌉ (-5)⋅ THENA (RewriteWithILF (-5) THEN Auto))
   THEN (Subst ⌈msg-authentic(info(e4)) ~ ff⌉ (-1)⋅ THENA (RewriteWithILF (-1) THEN Auto))

   THEN (Subst ⌈make-msg-interface(0;loc(e3);mk-msg(ff;``pv11_p1 p2a``;<l1, b, s, p1>)) ~ pv11_p1_p2a'send(Cmd;f) loc(e3\000C) <l1, b, s, p1>⌉ (-5)⋅

THENA (RepUR ``pv11_p1_p2a'send mk-msg-interface make-Msg mk-msg`` 0 THEN Auto)
)

   THEN (Subst ⌈make-msg-interface(0;loc(e4);mk-msg(ff;``pv11_p1 p2a``;<l2, b, s, p2>)) ~ pv11_p1_p2a'send(Cmd;f) loc(e4\000C) <l2, b, s, p2>⌉ (-1)⋅

THENA (RepUR ``pv11_p1_p2a'send mk-msg-interface make-Msg mk-msg`` 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

 (∀e1:E. ∀bnum1,bnum2:pv11_p1_Ballot_Num(). ∀accepted1,accepted2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.

            ∨ (pv11_p1_p2a'send(Cmd;f) i1 <l, b, s, p1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)

              ∧ pv11_p1_p2a'send(Cmd;f) i2 <l, b, s, p2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e3)))

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. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 18. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 19. e2 : E@i 20. \mforall{}e3:E ((e3 < e2) {}\mRightarrow{} (\mforall{}e1:E. \mforall{}bnum1,bnum2:pv11\_p1\_Ballot\_Num(). \mforall{}accepted1,accepted2:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List. \mforall{}i1,i2,l:Id. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}s:\mBbbZ{}. \mforall{}p1,p2:Cmd. (((<bnum1, accepted1> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e1\000C) \mwedge{} <bnum2, accepted2> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e3) \mwedge{} (<b, s, p1> \mmember{} accepted1) \mwedge{} (<b, s, p2> \mmember{} accepted2)) \mvee{} (pv11\_p1\_p2a'send(Cmd;f) i1 <l, b, s, p1> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$\mbackslash{}ff5\000Cf{uid}$;reps;f)(e1) \mwedge{} pv11\_p1\_p2a'send(Cmd;f) i2 <l, b, s, p2> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$\mbackslash{}ff\000C5f{uid}$;reps;f)(e3))) {}\mRightarrow{} (p1 = p2)))) 21. e1 : E@i 22. bnum1 : pv11\_p1\_Ballot\_Num()@i 23. bnum2 : pv11\_p1\_Ballot\_Num()@i 24. accepted1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 25. accepted2 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 26. i1 : Id@i 27. i2 : Id@i 28. l : Id@i 29. b : pv11\_p1\_Ballot\_Num()@i 30. s : \mBbbZ{}@i 31. p1 : Cmd@i 32. p2 : Cmd@i 33. <bnum1, accepted1> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e1)@i 34. <bnum2, accepted2> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e2)@i 35. (<b, s, p1> \mmember{} accepted1)@i 36. (<b, s, p2> \mmember{} accepted2)@i 37. e3 : E 38. l1 : Id 39. e3 \mleq{}loc e1 40. <l1, b, s, p1> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e3) 41. e4 : E 42. l2 : Id 43. e4 \mleq{}loc e2 44. <l2, b, s, p2> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e4) 45. e' : E 46. delay : \mBbbZ{} 47. (e' < e3) 48. make-msg-interface(delay;loc(e3);mk-msg(msg-authentic(info(e3));``pv11\_p1 p2a``;<l1, b, s, p1>))\000C \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}$;reps;f)(e') 49. e5 : E 50. d1 : \mBbbZ{} 51. (e5 < e4) 52. make-msg-interface(d1;loc(e4);mk-msg(msg-authentic(info(e4));``pv11\_p1 p2a``;<l2, b, s, p2>)) \mmember{} \000Cpv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}$;reps;f)(e5) \mvdash{} p1 = p2 By Latex: ((Subst \mkleeneopen{}delay \msim{} 0\mkleeneclose{} (-5)\mcdot{} THENA (RewriteWithILF (-5) THEN Auto)) THEN (Subst \mkleeneopen{}d1 \msim{} 0\mkleeneclose{} (-1)\mcdot{} THENA (RewriteWithILF (-1) THEN Auto)) THEN (Subst \mkleeneopen{}msg-authentic(info(e3)) \msim{} ff\mkleeneclose{} (-5)\mcdot{} THENA (RewriteWithILF (-5) THEN Auto)) THEN (Subst \mkleeneopen{}msg-authentic(info(e4)) \msim{} ff\mkleeneclose{} (-1)\mcdot{} THENA (RewriteWithILF (-1) THEN Auto)) THEN (Subst \mkleeneopen{}make-msg-interface(0;loc(e3);mk-msg(ff;``pv11\_p1 p2a``;<l1, b, s, p1>)) \msim{} pv11\_p1\_p2a'\000Csend(Cmd;f) loc(e3) <l1, b, s, p1>\mkleeneclose{} (-5)\mcdot{} THENA (RepUR ``pv11\_p1\_p2a'send mk-msg-interface make-Msg mk-msg`` 0 THEN Auto) ) THEN (Subst \mkleeneopen{}make-msg-interface(0;loc(e4);mk-msg(ff;``pv11\_p1 p2a``;<l2, b, s, p2>)) \msim{} pv11\_p1\_p2a'\000Csend(Cmd;f) loc(e4) <l2, b, s, p2>\mkleeneclose{} (-1)\mcdot{} THENA (RepUR ``pv11\_p1\_p2a'send mk-msg-interface make-Msg mk-msg`` 0 THEN Auto) )) Home Index