Proof of Nuprl Lemma : pv11_p1_A4

Step * 1 2 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. loc(e1) ↓∈ ldrs
34. i1 ↓∈ accpts
35. 0 = 0 ∈ ℤ
36. ff = pv11_p1_init_active()
37. l = loc(e1) ∈ Id
38. header(e1) = [propose] ∈ Name
39. has-es-info-type(es;e1;f;ℤ × Cmd)
40. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))))
41. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e1))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))))
42. b = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))) ∈ pv11_p1_Ballot_Num()
43. <s, p1> = msgval(e1) ∈ (ℤ × Cmd)
44. loc(e2) ↓∈ ldrs
45. i2 ↓∈ accpts
46. 0 = 0 ∈ ℤ
47. ff = pv11_p1_init_active()
48. l = loc(e2) ∈ Id
49. header(e2) = [propose] ∈ Name
50. has-es-info-type(es;e2;f;ℤ × Cmd)
51. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2))))
52. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e2))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2)))))
53. b = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2))) ∈ pv11_p1_Ballot_Num()
54. <s, p2> = msgval(e2) ∈ (ℤ × Cmd)
55. loc(e1) = loc(e2) ∈ Id
⊢ p1 = p2 ∈ Cmd
BY
{ UseLoclTri ⌈es⌉⌈e1⌉⌈e2⌉⋅ }

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)))

⇒ (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. loc(e1) ↓∈ ldrs
34. i1 ↓∈ accpts
35. 0 = 0 ∈ ℤ
36. ff = pv11_p1_init_active()
37. l = loc(e1) ∈ Id
38. header(e1) = [propose] ∈ Name
39. has-es-info-type(es;e1;f;ℤ × Cmd)
40. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))))
41. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e1))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))))
42. b = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))) ∈ pv11_p1_Ballot_Num()
43. <s, p1> = msgval(e1) ∈ (ℤ × Cmd)
44. loc(e2) ↓∈ ldrs
45. i2 ↓∈ accpts
46. 0 = 0 ∈ ℤ
47. ff = pv11_p1_init_active()
48. l = loc(e2) ∈ Id
49. header(e2) = [propose] ∈ Name
50. has-es-info-type(es;e2;f;ℤ × Cmd)
51. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2))))
52. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e2))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2)))))
53. b = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2))) ∈ pv11_p1_Ballot_Num()
54. <s, p2> = msgval(e2) ∈ (ℤ × Cmd)
55. loc(e1) = loc(e2) ∈ Id
56. (e1 <loc e2)
⊢ p1 = p2 ∈ Cmd

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

 (∀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. loc(e1) \mdownarrow{}\mmember{} ldrs 34. i1 \mdownarrow{}\mmember{} accpts 35. 0 = 0 36. ff = pv11\_p1\_init\_active() 37. l = loc(e1) 38. header(e1) = [propose] 39. has-es-info-type(es;e1;f;\mBbbZ{} \mtimes{} Cmd) 40. \muparrow{}(fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)))) 41. \mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) (fst(msgval(e1))) (snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1))))) 42. b = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1))) 43. <s, p1> = msgval(e1) 44. loc(e2) \mdownarrow{}\mmember{} ldrs 45. i2 \mdownarrow{}\mmember{} accpts 46. 0 = 0 47. ff = pv11\_p1\_init\_active() 48. l = loc(e2) 49. header(e2) = [propose] 50. has-es-info-type(es;e2;f;\mBbbZ{} \mtimes{} Cmd) 51. \muparrow{}(fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e2)))) 52. \mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) (fst(msgval(e2))) (snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e2))))) 53. b = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e2))) 54. <s, p2> = msgval(e2) 55. loc(e1) = loc(e2) \mvdash{} p1 = p2 By Latex: UseLoclTri \mkleeneopen{}es\mkleeneclose{}\mkleeneopen{}e1\mkleeneclose{}\mkleeneopen{}e2\mkleeneclose{}\mcdot{} Home Index