Proof of Nuprl Lemma : pv11_p1_A4

Step * 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
⊢ p1 = p2 ∈ Cmd
BY

{ ((InstLemma `pv11_p1_acc_p2a` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈e1⌉;⌈ldrs_uid⌉;⌈<bnum1, accepted1>⌉;⌈b⌉;⌈s⌉;⌈p1⌉]⋅ THENA Auto)

   THEN (InstLemma `pv11_p1_acc_p2a` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈e2⌉;⌈ldrs_uid⌉;⌈<bnum2, accepted2>⌉;⌈b⌉;⌈s⌉;⌈p2⌉]⋅ THENA Auto)

   THEN All Reduce
   THEN RepeatFor 2 ((D (-2) THENA Auto))
   THEN SquashExRepD
   THEN RenameVar `l0' (-7)
   THEN RenameVar `l2' (-3)
   THEN RenameVar `l1' (-7)
   THEN RenameVar `e4' (-4)
   THEN RenameVar `e3' (-8)
   THEN RepeatFor 2 ((UseMessageConstraint' (-5) THENA 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)))

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

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 \mvdash{} p1 = p2 By Latex: ((InstLemma `pv11\_p1\_acc\_p2a` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e1\mkleeneclose{};\mkleeneopen{}ldrs$_{uid}$\mkleeneclose{};\mkleeneopen{}<bnum1, accep\000Cted1>\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}s\mkleeneclose{};\mkleeneopen{}p1\mkleeneclose{}]\mcdot{} THENA Auto ) THEN (InstLemma `pv11\_p1\_acc\_p2a` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{};\mkleeneopen{}ldrs$_{uid}$\mkleeneclose{};\mkleeneopen{}<bnum2, \000Caccepted2>\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}s\mkleeneclose{};\mkleeneopen{}p2\mkleeneclose{}] \mcdot{} THENA Auto ) THEN All Reduce THEN RepeatFor 2 ((D (-2) THENA Auto)) THEN SquashExRepD THEN RenameVar `l0' (-7) THEN RenameVar `l2' (-3) THEN RenameVar `l1' (-7) THEN RenameVar `e4' (-4) THEN RenameVar `e3' (-8) THEN RepeatFor 2 ((UseMessageConstraint' (-5) THENA Auto))) Home Index