Step
*
1
1
1
of Lemma
new_23_sig_progress-step4
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. c : Cmd
21. faulty : bag(Id)
22. #(faulty) ≤ flrs@i
23. ∀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
24. bag-no-repeats(Id;reps)@i
25. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
26. loc(e) ↓∈ reps@i
27. ¬loc(e) ↓∈ faulty@i
28. <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)@i
29. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
30. ∀i:Id
(i ↓∈ reps
⇒ (↓∃e':E. ∃c':Cmd. ((loc(e') = i ∈ Id) ∧ mk-msg-interface(loc(e);make-Msg(``new_23_sig vote``;<<<n, 0>, c'>, i>)\000C) ∈ new_23_sig_main()(e'))))
31. i : Id@i
32. i ↓∈ reps
33. ¬i ↓∈ faulty
34. e' : E
35. c' : Cmd
36. loc(e') = i ∈ Id
37. mk-msg-interface(loc(e);make-Msg(``new_23_sig vote``;<<<n, 0>, c'>, i>)) ∈ new_23_sig_main()(e')
38. e'@0 : E
39. delay : ℤ
40. (e' < e'@0)
41. 0 = delay ∈ ℤ
42. loc(e) = loc(e'@0) ∈ Id
43. make-Msg(``new_23_sig vote``;<<<n, 0>, c'>, i>) = info(e'@0) ∈ Message(f)
44. loc(e'@0) = loc(e) ∈ Id
⊢ info(e'@0) = mk-msg(msg-authentic(info(e'@0));``new_23_sig vote``;<<<n, 0>, c'>, i>) ∈ Message(f)
BY
{ (Subst' msg-authentic(info(e'@0)) = ff 0 THENA (RevHypSubst' (-2) 0 THEN Auto)) }
1
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. c : Cmd
21. faulty : bag(Id)
22. #(faulty) ≤ flrs@i
23. ∀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
24. bag-no-repeats(Id;reps)@i
25. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
26. loc(e) ↓∈ reps@i
27. ¬loc(e) ↓∈ faulty@i
28. <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)@i
29. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
30. ∀i:Id
(i ↓∈ reps
⇒ (↓∃e':E. ∃c':Cmd. ((loc(e') = i ∈ Id) ∧ mk-msg-interface(loc(e);make-Msg(``new_23_sig vote``;<<<n, 0>, c'>, i>)\000C) ∈ new_23_sig_main()(e'))))
31. i : Id@i
32. i ↓∈ reps
33. ¬i ↓∈ faulty
34. e' : E
35. c' : Cmd
36. loc(e') = i ∈ Id
37. mk-msg-interface(loc(e);make-Msg(``new_23_sig vote``;<<<n, 0>, c'>, i>)) ∈ new_23_sig_main()(e')
38. e'@0 : E
39. delay : ℤ
40. (e' < e'@0)
41. 0 = delay ∈ ℤ
42. loc(e) = loc(e'@0) ∈ Id
43. make-Msg(``new_23_sig vote``;<<<n, 0>, c'>, i>) = info(e'@0) ∈ Message(f)
44. loc(e'@0) = loc(e) ∈ Id
⊢ info(e'@0) = mk-msg(ff;``new_23_sig vote``;<<<n, 0>, c'>, i>) ∈ Message(f)
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. c : Cmd
21. faulty : bag(Id)
22. \#(faulty) \mleq{} flrs@i
23. \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
24. bag-no-repeats(Id;reps)@i
25. \#(reps) = ((coeff * flrs) + flrs + 1)@i
26. loc(e) \mdownarrow{}\mmember{} reps@i
27. \mneg{}loc(e) \mdownarrow{}\mmember{} faulty@i
28. <n, c> \mmember{} new\_23\_sig\_Proposal(Cmd;notify;propose;f)(e)@i
29. \mneg{}\muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
30. \mforall{}i:Id
(i \mdownarrow{}\mmember{} reps
{}\mRightarrow{} (\mdownarrow{}\mexists{}e':E
\mexists{}c':Cmd
((loc(e') = i) \mwedge{} mk-msg-interface(loc(e);make-Msg(``new\_23\_sig vote``;<<<n, 0>, c'>, i>\000C)) \mmember{} new\_23\_sig\_main()(e'))))
31. i : Id@i
32. i \mdownarrow{}\mmember{} reps
33. \mneg{}i \mdownarrow{}\mmember{} faulty
34. e' : E
35. c' : Cmd
36. loc(e') = i
37. mk-msg-interface(loc(e);make-Msg(``new\_23\_sig vote``;<<<n, 0>, c'>, i>)) \mmember{} new\_23\_sig\_main()(e')
38. e'@0 : E
39. delay : \mBbbZ{}
40. (e' < e'@0)
41. 0 = delay
42. loc(e) = loc(e'@0)
43. make-Msg(``new\_23\_sig vote``;<<<n, 0>, c'>, i>) = info(e'@0)
44. loc(e'@0) = loc(e)
\mvdash{} info(e'@0) = mk-msg(msg-authentic(info(e'@0));``new\_23\_sig vote``;<<<n, 0>, c'>, i>)
By
Latex:
(Subst' msg-authentic(info(e'@0)) = ff 0 THENA (RevHypSubst' (-2) 0 THEN Auto))
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