Step
*
1
1
1
1
of Lemma
new_23_sig_agreement
.....assertion.....
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. es : EO+(Message(f))
12. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. bag-no-repeats(Id;reps)@i
14. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
15. (f propose) = (ℤ × Cmd) ∈ Type
16. (f notify) = (ℤ × Cmd) ∈ Type
17. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
18. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
19. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
20. f ∈ Name ─→ Type
21. e1 : E@i
22. e2 : E@i
23. c1 : Cmd@i
24. n : ℤ@i
25. c2 : Cmd@i
26. <n, c1> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e1)@i
27. <n, c2> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e2)@i
28. x1 : Id
29. r1 : ℕ
30. L1 : Id List
31. no_repeats(Id;L1)
32. ||L1|| = ((coeff * flrs) + 1) ∈ ℤ
33. (∀vtr∈L1.↓∃e':E. mk-msg-interface(x1;make-Msg(``new_23_sig vote``;<<<n, r1>, c1>, vtr>)) ∈ new_23_sig_main()(e'))
34. x : Id
35. rnd : ℕ
36. L : Id List
37. no_repeats(Id;L)
38. ||L|| = ((coeff * flrs) + 1) ∈ ℤ
39. (∀vtr∈L.↓∃e':E. mk-msg-interface(x;make-Msg(``new_23_sig vote``;<<<n, rnd>, c2>, vtr>)) ∈ new_23_sig_main()(e'))
40. r1 = rnd ∈ ℤ
41. ¬(∃vtr∈L1. (vtr ∈ L))
⊢ ∃b:bag(Id). ((#(b) = (((2 * coeff) * flrs) + 2) ∈ ℤ) ∧ bag-no-repeats(Id;b) ∧ (∀x:Id. (x ↓∈ b ⇒ x ↓∈ reps)))
BY
{ (With ⌈L @ L1⌉ (D 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. es : EO+(Message(f))
12. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. bag-no-repeats(Id;reps)@i
14. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
15. (f propose) = (ℤ × Cmd) ∈ Type
16. (f notify) = (ℤ × Cmd) ∈ Type
17. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
18. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
19. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
20. f ∈ Name ─→ Type
21. e1 : E@i
22. e2 : E@i
23. c1 : Cmd@i
24. n : ℤ@i
25. c2 : Cmd@i
26. <n, c1> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e1)@i
27. <n, c2> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e2)@i
28. x1 : Id
29. r1 : ℕ
30. L1 : Id List
31. no_repeats(Id;L1)
32. ||L1|| = ((coeff * flrs) + 1) ∈ ℤ
33. (∀vtr∈L1.↓∃e':E. mk-msg-interface(x1;make-Msg(``new_23_sig vote``;<<<n, r1>, c1>, vtr>)) ∈ new_23_sig_main()(e'))
34. x : Id
35. rnd : ℕ
36. L : Id List
37. no_repeats(Id;L)
38. ||L|| = ((coeff * flrs) + 1) ∈ ℤ
39. (∀vtr∈L.↓∃e':E. mk-msg-interface(x;make-Msg(``new_23_sig vote``;<<<n, rnd>, c2>, vtr>)) ∈ new_23_sig_main()(e'))
40. r1 = rnd ∈ ℤ
41. ¬(∃vtr∈L1. (vtr ∈ L))
⊢ #(L @ L1) = (((2 * coeff) * flrs) + 2) ∈ ℤ
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. es : EO+(Message(f))
12. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. bag-no-repeats(Id;reps)@i
14. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
15. (f propose) = (ℤ × Cmd) ∈ Type
16. (f notify) = (ℤ × Cmd) ∈ Type
17. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
18. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
19. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
20. f ∈ Name ─→ Type
21. e1 : E@i
22. e2 : E@i
23. c1 : Cmd@i
24. n : ℤ@i
25. c2 : Cmd@i
26. <n, c1> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e1)@i
27. <n, c2> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e2)@i
28. x1 : Id
29. r1 : ℕ
30. L1 : Id List
31. no_repeats(Id;L1)
32. ||L1|| = ((coeff * flrs) + 1) ∈ ℤ
33. (∀vtr∈L1.↓∃e':E. mk-msg-interface(x1;make-Msg(``new_23_sig vote``;<<<n, r1>, c1>, vtr>)) ∈ new_23_sig_main()(e'))
34. x : Id
35. rnd : ℕ
36. L : Id List
37. no_repeats(Id;L)
38. ||L|| = ((coeff * flrs) + 1) ∈ ℤ
39. (∀vtr∈L.↓∃e':E. mk-msg-interface(x;make-Msg(``new_23_sig vote``;<<<n, rnd>, c2>, vtr>)) ∈ new_23_sig_main()(e'))
40. r1 = rnd ∈ ℤ
41. ¬(∃vtr∈L1. (vtr ∈ L))
42. #(L @ L1) = (((2 * coeff) * flrs) + 2) ∈ ℤ
⊢ bag-no-repeats(Id;L @ L1)
3
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. es : EO+(Message(f))
12. new_23_sig_message-constraint{i:l}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. bag-no-repeats(Id;reps)@i
14. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
15. (f propose) = (ℤ × Cmd) ∈ Type
16. (f notify) = (ℤ × Cmd) ∈ Type
17. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
18. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
19. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
20. f ∈ Name ─→ Type
21. e1 : E@i
22. e2 : E@i
23. c1 : Cmd@i
24. n : ℤ@i
25. c2 : Cmd@i
26. <n, c1> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e1)@i
27. <n, c2> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e2)@i
28. x1 : Id
29. r1 : ℕ
30. L1 : Id List
31. no_repeats(Id;L1)
32. ||L1|| = ((coeff * flrs) + 1) ∈ ℤ
33. (∀vtr∈L1.↓∃e':E. mk-msg-interface(x1;make-Msg(``new_23_sig vote``;<<<n, r1>, c1>, vtr>)) ∈ new_23_sig_main()(e'))
34. x : Id
35. rnd : ℕ
36. L : Id List
37. no_repeats(Id;L)
38. ||L|| = ((coeff * flrs) + 1) ∈ ℤ
39. (∀vtr∈L.↓∃e':E. mk-msg-interface(x;make-Msg(``new_23_sig vote``;<<<n, rnd>, c2>, vtr>)) ∈ new_23_sig_main()(e'))
40. r1 = rnd ∈ ℤ
41. ¬(∃vtr∈L1. (vtr ∈ L))
42. #(L @ L1) = (((2 * coeff) * flrs) + 2) ∈ ℤ
43. bag-no-repeats(Id;L @ L1)
44. x2 : Id@i
45. x2 ↓∈ L @ L1@i
⊢ x2 ↓∈ reps
Latex:
Latex:
.....assertion.....
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. es : EO+(Message(f))
12. new\_23\_sig\_message-constraint\{i:l\}(Cmd;clients;eq;coeff;flrs;notify;propose;reps;slots;f)@i'
13. bag-no-repeats(Id;reps)@i
14. \#(reps) = ((coeff * flrs) + flrs + 1)@i
15. (f propose) = (\mBbbZ{} \mtimes{} Cmd)
16. (f notify) = (\mBbbZ{} \mtimes{} Cmd)
17. (f ``new\_23\_sig decided``) = (\mBbbZ{} \mtimes{} Cmd)
18. (f ``new\_23\_sig retry``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd)
19. (f ``new\_23\_sig vote``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
20. f \mmember{} Name {}\mrightarrow{} Type
21. e1 : E@i
22. e2 : E@i
23. c1 : Cmd@i
24. n : \mBbbZ{}@i
25. c2 : Cmd@i
26. <n, c1> \mmember{} new\_23\_sig\_decided'base(Cmd;notify;propose;f)(e1)@i
27. <n, c2> \mmember{} new\_23\_sig\_decided'base(Cmd;notify;propose;f)(e2)@i
28. x1 : Id
29. r1 : \mBbbN{}
30. L1 : Id List
31. no\_repeats(Id;L1)
32. ||L1|| = ((coeff * flrs) + 1)
33. (\mforall{}vtr\mmember{}L1.\mdownarrow{}\mexists{}e':E. mk-msg-interface(x1;make-Msg(``new\_23\_sig vote``;<<<n, r1>, c1>, vtr>)) \mmember{} new\_2\000C3\_sig\_main()(e'))
34. x : Id
35. rnd : \mBbbN{}
36. L : Id List
37. no\_repeats(Id;L)
38. ||L|| = ((coeff * flrs) + 1)
39. (\mforall{}vtr\mmember{}L.\mdownarrow{}\mexists{}e':E. mk-msg-interface(x;make-Msg(``new\_23\_sig vote``;<<<n, rnd>, c2>, vtr>)) \mmember{} new\_23\000C\_sig\_main()(e'))
40. r1 = rnd
41. \mneg{}(\mexists{}vtr\mmember{}L1. (vtr \mmember{} L))
\mvdash{} \mexists{}b:bag(Id)
((\#(b) = (((2 * coeff) * flrs) + 2)) \mwedge{} bag-no-repeats(Id;b) \mwedge{} (\mforall{}x:Id. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} x \mdownarrow{}\mmember{} reps)))
By
Latex:
(With \mkleeneopen{}L @ L1\mkleeneclose{} (D 0)\mcdot{} THEN Auto)\mcdot{}
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