Proof of Nuprl Lemma : pv11_p1_acc_rcv

Step * 1 4 1 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

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

BY
{ (Duplicate (-1)
   THEN UseClassRel (-1)
   THEN (UseMessageConstraint' (-3) THENA Auto)
   THEN Try (Complete ((Repeat ((RWO "pair-eta<" 0 THENA Auto)) THEN Auto)))
   THEN (Subst ⌈delay ~ 0⌉ (-1)⋅ THENA (RewriteWithILF (-1) THEN Auto))
   THEN (Subst ⌈es-info-auth(es;e1) ~ ff⌉ (-1)⋅ THENA (RewriteWithILF (-1) THEN Auto))
   THEN (Subst ⌈make-msg-interface(0;loc(e1);mk-msg(ff;``pv11_p1 p2a``;<fst(msgval(e1))

                                                                       , fst(snd(msgval(e1)))

                                                                       , fst(snd(snd(msgval(e1))))

                                                                       , snd(snd(snd(msgval(e1))))>)) ~ pv11_p1_p2a'send\000C(Cmd;f) loc(e1) msgval(e1)⌉ (-1)⋅

         THENA (RepUR ``pv11_p1_p2a'send mk-msg-interface mk-msg make-Msg`` 0
                THEN Auto
                THEN Repeat ((RWO "pair-eta<" 0 THEN Auto)))
         )
   THEN DProdsAndUnions

   THEN (InstLemma `pv11_p1_A4_C1` [⌈Cmd⌉;⌈f⌉;⌈es⌉;⌈e'⌉;⌈e2⌉;⌈accpts⌉;⌈ldrs⌉;⌈ldrs_uid⌉;⌈reps⌉;⌈b⌉;⌈b⌉;⌈[]⌉;⌈[]⌉;⌈i⌉;

         ⌈loc(e1)⌉;⌈l1⌉;⌈z1⌉;⌈z3⌉;⌈snd(snd(snd(msgval(e))))⌉;⌈z4⌉]⋅
         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

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. z1 : pv11_p1_Ballot_Num()
36. z3 : ℤ
37. z4 : Cmd
38. (<z1, z3, z4> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))))
39. ↑(pv11_p1_same_pvalue(Cmd) <fst(snd(msgval(e))), snd(snd(msgval(e)))> <z1, z3, z4>)
40. e1 : E
41. l1 : Id
42. e1 ≤loc pred(e)
43. <l1, z1, z3, z4> ∈ pv11_p1_p2a'base(Cmd;f)(e1)
44. header(e1) = ``pv11_p1 p2a`` ∈ Name
45. has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
46. <l1, z1, z3, z4> = msgval(e1) ∈ (Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
47. ∀e:E
      ((header(e) ∈ pv11_p1_headers_no_inputs())
      ⇒ (↓∃e':E
            ∃delay:ℤ

             ((e' < e) ∧ make-msg-interface(delay;loc(e);info(e)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e'))))

48. e2 : E
49. delay : ℤ
50. (e2 < e1)
51. pv11_p1_p2a'send(Cmd;f) loc(e1) msgval(e1) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2)@i
52. msgval(e) ∈ Id × pv11_p1_Ballot_Num() × ℤ × Cmd
⊢ (<b, []> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e')
∧ <b, []> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e2)
∧ (<z1, z3, snd(snd(snd(msgval(e))))> ∈ [])
∧ (<z1, z3, z4> ∈ []))

∨ (pv11_p1_p2a'send(Cmd;f) i <l1, z1, z3, snd(snd(snd(msgval(e))))> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e')

∧ pv11_p1_p2a'send(Cmd;f) loc(e1) <l1, z1, z3, z4> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2))

2
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. z1 : pv11_p1_Ballot_Num()
36. z3 : ℤ
37. z4 : Cmd
38. (<z1, z3, z4> ∈ snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))))
39. ↑(pv11_p1_same_pvalue(Cmd) <fst(snd(msgval(e))), snd(snd(msgval(e)))> <z1, z3, z4>)
40. e1 : E
41. l1 : Id
42. e1 ≤loc pred(e)
43. <l1, z1, z3, z4> ∈ pv11_p1_p2a'base(Cmd;f)(e1)
44. header(e1) = ``pv11_p1 p2a`` ∈ Name
45. has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
46. <l1, z1, z3, z4> = msgval(e1) ∈ (Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
47. ∀e:E
      ((header(e) ∈ pv11_p1_headers_no_inputs())
      ⇒ (↓∃e':E
            ∃delay:ℤ

             ((e' < e) ∧ make-msg-interface(delay;loc(e);info(e)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e'))))

48. e2 : E
49. delay : ℤ
50. (e2 < e1)
51. pv11_p1_p2a'send(Cmd;f) loc(e1) msgval(e1) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e2)@i
52. (snd(snd(snd(msgval(e))))) = z4 ∈ Cmd

⊢ ↓(<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. e' : E@i 28. i : Id@i 29. l : Id@i 30. pv11\_p1\_p2a'send(Cmd;f) i <l, snd(msgval(e))> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{\000Cuid}$;reps;f)(e')@i 31. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f) 32. \muparrow{}e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f) 33. \muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) (fst(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$\000C_{uid}$;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))))@i 35. z : pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd 36. (z \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e)))) 37. \muparrow{}(pv11\_p1\_same\_pvalue(Cmd) <fst(snd(msgval(e))), snd(snd(msgval(e)))> z) 38. e1 : E 39. l1 : Id 40. e1 \mleq{}loc pred(e) 41. <l1, z> \mmember{} pv11\_p1\_p2a'base(Cmd;f)(e1) \mvdash{} \mdownarrow{}(<fst(snd(msgval(e))), snd(snd(msgval(e)))> \mmember{} snd(pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e)))) \mvee{} False By Latex: (Duplicate (-1) THEN UseClassRel (-1) THEN (UseMessageConstraint' (-3) THENA Auto) THEN Try (Complete ((Repeat ((RWO "pair-eta<" 0 THENA Auto)) THEN Auto))) THEN (Subst \mkleeneopen{}delay \msim{} 0\mkleeneclose{} (-1)\mcdot{} THENA (RewriteWithILF (-1) THEN Auto)) THEN (Subst \mkleeneopen{}es-info-auth(es;e1) \msim{} ff\mkleeneclose{} (-1)\mcdot{} THENA (RewriteWithILF (-1) THEN Auto)) THEN (Subst \mkleeneopen{}make-msg-interface(0;loc(e1);mk-msg(ff;``pv11\_p1 p2a``;<fst(msgval(e1)) , fst(snd(msgval(e1))) , fst(snd(snd(msgval(e1)))) , snd(snd(snd(msgval(e1))))>)) \000C\msim{} pv11\_p1\_p2a'send(Cmd;f) loc(e1) msgval(e1)\mkleeneclose{} (-1)\mcdot{} THENA (RepUR ``pv11\_p1\_p2a'send mk-msg-interface mk-msg make-Msg`` 0 THEN Auto THEN Repeat ((RWO "pair-eta<" 0 THEN Auto))) ) THEN DProdsAndUnions THEN (InstLemma `pv11\_p1\_A4\_C1` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e'\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{};\mkleeneopen{}accpts\mkleeneclose{};\mkleeneopen{}ldrs\mkleeneclose{};\mkleeneopen{}ldrs$_{uid\mbackslash{}f\000Cf7d$\mkleeneclose{};\mkleeneopen{}reps\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{}; \mkleeneopen{}[]\mkleeneclose{};\mkleeneopen{}[]\mkleeneclose{};\mkleeneopen{}i\mkleeneclose{};\mkleeneopen{}loc(e1)\mkleeneclose{};\mkleeneopen{}l1\mkleeneclose{};\mkleeneopen{}z1\mkleeneclose{};\mkleeneopen{}z3\mkleeneclose{};\mkleeneopen{}snd(snd(snd(msgval(e))))\mkleeneclose{};\mkleeneopen{}z4\mkleeneclose{}]\mcdot{} THENA Auto )) Home Index