Proof of Nuprl Lemma : new_23_sig

Step * 1 1 of Lemma new_23_sig_progress2-step2

1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : {2...}
6. flrs : ℕ
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig{i:l}(ℤ)
10. f : new_23_sig_headers_type{i:l}(Cmd;notify;propose)
11. (f propose) = (ℤ × Cmd) ∈ Type
12. (f notify) = (ℤ × Cmd) ∈ Type
13. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
14. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
15. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
16. f ∈ Name ─→ Type
17. es : EO+(Message(f))
18. e : E
19. n : ℤ
20. r : ℤ
21. c : Cmd
22. faulty : bag(Id)
23. e' : E
24. c' : Cmd
25. #(faulty) ≤ flrs@i
26. ∀e:E. ∀dmsg:Interface.
      (dmsg ∈ new_23_sig_main()(e)
      ⇒ (¬loc(e) ↓∈ faulty)
      ⇒ loc(e) ↓∈ reps
      ⇒

 (↓∃e':E. ∃delay:ℤ. ((e < e') ∧ (dmsg = make-msg-interface(delay;loc(e');info(e')) ∈ Interface))))@i

27. bag-no-repeats(Id;reps)@i
28. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
29. loc(e) ↓∈ reps@i
30. ¬loc(e) ↓∈ faulty@i
31. e' ≤loc e @i
32. <n, c'> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e')@i
33. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))@i
34. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) n between e' and e)@i
35. <<n, r>, c> ∈ new_23_sig_RoundInfo(Cmd;notify;propose;f)(e)@i
36. new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r@i
37. ∀i:Id. (i ↓∈ reps ⇒ make-msg-interface(0;i;make-Msg(``new_23_sig vote``;<<<n, r>, c>, loc(e)>)) ∈ new_23_sig_main()\000C(e))
38. i : Id@i
39. i ↓∈ reps@i
40. make-msg-interface(0;i;make-Msg(``new_23_sig vote``;<<<n, r>, c>, loc(e)>)) ∈ new_23_sig_main()(e)
41. e1 : E
42. delay : ℤ
43. (e < e1)
44. 0 = delay ∈ ℤ
45. i = loc(e1) ∈ Id
46. make-Msg(``new_23_sig vote``;<<<n, r>, c>, loc(e)>) = info(e1) ∈ Message(f)
47. (e < e1)
48. loc(e1) = i ∈ Id
⊢ info(e1) = mk-msg(msg-authentic(info(e1));``new_23_sig vote``;<<<n, r>, c>, loc(e)>) ∈ Message(f)
BY
{ (Subst' msg-authentic(info(e1)) = ff 0 THEN Auto) }

1
.....equality.....
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : {2...}
6. flrs : ℕ
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig{i:l}(ℤ)
10. f : new_23_sig_headers_type{i:l}(Cmd;notify;propose)
11. (f propose) = (ℤ × Cmd) ∈ Type
12. (f notify) = (ℤ × Cmd) ∈ Type
13. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
14. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
15. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
16. f ∈ Name ─→ Type
17. es : EO+(Message(f))
18. e : E
19. n : ℤ
20. r : ℤ
21. c : Cmd
22. faulty : bag(Id)
23. e' : E
24. c' : Cmd
25. #(faulty) ≤ flrs@i
26. ∀e:E. ∀dmsg:Interface.
      (dmsg ∈ new_23_sig_main()(e)
      ⇒ (¬loc(e) ↓∈ faulty)
      ⇒ loc(e) ↓∈ reps
      ⇒

 (↓∃e':E. ∃delay:ℤ. ((e < e') ∧ (dmsg = make-msg-interface(delay;loc(e');info(e')) ∈ Interface))))@i

27. bag-no-repeats(Id;reps)@i
28. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
29. loc(e) ↓∈ reps@i
30. ¬loc(e) ↓∈ faulty@i
31. e' ≤loc e @i
32. <n, c'> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e')@i
33. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))@i
34. (no new_23_sig_Notify(Cmd;clients;notify;propose;f) n between e' and e)@i
35. <<n, r>, c> ∈ new_23_sig_RoundInfo(Cmd;notify;propose;f)(e)@i
36. new_23_sig_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r@i
37. ∀i:Id. (i ↓∈ reps ⇒ make-msg-interface(0;i;make-Msg(``new_23_sig vote``;<<<n, r>, c>, loc(e)>)) ∈ new_23_sig_main()\000C(e))
38. i : Id@i
39. i ↓∈ reps@i
40. make-msg-interface(0;i;make-Msg(``new_23_sig vote``;<<<n, r>, c>, loc(e)>)) ∈ new_23_sig_main()(e)
41. e1 : E
42. delay : ℤ
43. (e < e1)
44. 0 = delay ∈ ℤ
45. i = loc(e1) ∈ Id
46. make-Msg(``new_23_sig vote``;<<<n, r>, c>, loc(e)>) = info(e1) ∈ Message(f)
47. (e < e1)
48. loc(e1) = i ∈ Id
⊢ msg-authentic(info(e1)) = ff

2
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : {2...}
6. flrs : ℕ
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig{i:l}(ℤ)
10. f : new_23_sig_headers_type{i:l}(Cmd;notify;propose)
11. (f propose) = (ℤ × Cmd) ∈ Type
12. (f notify) = (ℤ × Cmd) ∈ Type
13. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
14. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
15. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
16. f ∈ Name ─→ Type
17. es : EO+(Message(f))
18. e : E
19. n : ℤ
20. r : ℤ
21. c : Cmd
22. faulty : bag(Id)
23. e' : E
24. c' : Cmd
25. #(faulty) ≤ flrs@i
26. ∀e:E. ∀dmsg:Interface.
      (dmsg ∈ new_23_sig_main()(e)
      ⇒ (¬loc(e) ↓∈ faulty)
      ⇒ loc(e) ↓∈ reps
      ⇒

 (↓∃e':E. ∃delay:ℤ. ((e < e') ∧ (dmsg = make-msg-interface(delay;loc(e');info(e')) ∈ Interface))))@i

Latex:

Latex:
1. Cmd : \{T:Type| valueall-type(T)\} 2. eq : EqDecider(Cmd) 3. reps : bag(Id) 4. clients : bag(Id) 5. coeff : \{2...\} 6. flrs : \mBbbN{} 7. propose : Atom List 8. notify : Atom List 9. slots : set-sig\{i:l\}(\mBbbZ{}) 10. f : new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose) 11. (f propose) = (\mBbbZ{} \mtimes{} Cmd) 12. (f notify) = (\mBbbZ{} \mtimes{} Cmd) 13. (f ``new\_23\_sig decided``) = (\mBbbZ{} \mtimes{} Cmd) 14. (f ``new\_23\_sig retry``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd) 15. (f ``new\_23\_sig vote``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id) 16. f \mmember{} Name {}\mrightarrow{} Type 17. es : EO+(Message(f)) 18. e : E 19. n : \mBbbZ{} 20. r : \mBbbZ{} 21. c : Cmd 22. faulty : bag(Id) 23. e' : E 24. c' : Cmd 25. \#(faulty) \mleq{} flrs@i 26. \mforall{}e:E. \mforall{}dmsg:Interface. (dmsg \mmember{} new\_23\_sig\_main()(e) {}\mRightarrow{} (\mneg{}loc(e) \mdownarrow{}\mmember{} faulty) {}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} reps {}\mRightarrow{} (\mdownarrow{}\mexists{}e':E. \mexists{}delay:\mBbbZ{}. ((e < e') \mwedge{} (dmsg = make-msg-interface(delay;loc(e');info(e'))))))@i 27. bag-no-repeats(Id;reps)@i 28. \#(reps) = ((coeff * flrs) + flrs + 1)@i 29. loc(e) \mdownarrow{}\mmember{} reps@i 30. \mneg{}loc(e) \mdownarrow{}\mmember{} faulty@i 31. e' \mleq{}loc e @i 32. <n, c'> \mmember{} new\_23\_sig\_Proposal(Cmd;notify;propose;f)(e')@i 33. \mneg{}\muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e'))@i 34. (no new\_23\_sig\_Notify(Cmd;clients;notify;propose;f) n between e' and e)@i 35. <<n, r>, c> \mmember{} new\_23\_sig\_RoundInfo(Cmd;notify;propose;f)(e)@i 36. new\_23\_sig\_NewRoundsStateFun(Cmd;notify;propose;f;n;es.e';e) < r@i 37. \mforall{}i:Id. (i \mdownarrow{}\mmember{} reps {}\mRightarrow{} make-msg-interface(0;i;make-Msg(``new\_23\_sig vote``;<<<n, r>, c>, loc(e)>))\000C \mmember{} new\_23\_sig\_main()(e)) 38. i : Id@i 39. i \mdownarrow{}\mmember{} reps@i 40. make-msg-interface(0;i;make-Msg(``new\_23\_sig vote``;<<<n, r>, c>, loc(e)>)) \mmember{} new\_23\_sig\_main()(\000Ce) 41. e1 : E 42. delay : \mBbbZ{} 43. (e < e1) 44. 0 = delay 45. i = loc(e1) 46. make-Msg(``new\_23\_sig vote``;<<<n, r>, c>, loc(e)>) = info(e1) 47. (e < e1) 48. loc(e1) = i \mvdash{} info(e1) = mk-msg(msg-authentic(info(e1));``new\_23\_sig vote``;<<<n, r>, c>, loc(e)>) By Latex: (Subst' msg-authentic(info(e1)) = ff 0 THEN Auto) Home Index