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of Lemma
new_23_sig_progress-step10
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. msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)@i
23. bag-no-repeats(Id;reps)@i
24. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
25. loc(e) ↓∈ reps@i
26. ¬loc(e) ↓∈ faulty@i
27. <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)@i
28. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
29. e' : E
30. ((coeff * flrs) + 1)
= ||mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e'])||
∈ ℤ
31. e ≤loc e'
32. ↑new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0)
33. ¬↑(∃x∈[e, e'].new_23_sig_decided_with_id(Cmd;notify;propose;f;es;x;n))_b
34. 0 ≤ (coeff * flrs)
35. e ≤loc e'
36. ¬↑null(filter(λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);[e, e']))
37. ↑new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e';n;0)
38. has-es-info-type(es;e';f;ℤ × ℤ × Cmd × Id)
39. poss-maj(eq;rev(mapfilter(λe.(snd(fst(msgval(e))));
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e']));snd(fst(msgval(e')))) ∈ ℤ × Cmd
40. e ≤loc e'
41. (fst(poss-maj(eq;rev(mapfilter(λe.(snd(fst(msgval(e))));
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e']));snd(fst(msgval(e'))))))
= ((coeff * flrs) + 1)
∈ ℤ
42. loc(e') ↓∈ reps
43. e'@0 : E@i
44. (e'@0 <loc e')@i
45. e ≤loc e'@0 @i
46. w : Interface@i
47. f1 : (Cmd List × (Id List)) ─→ bag(Interface)
48. b1 : Cmd List
49. b2 : Id List
50. b7 : ℤ
51. b8 : ℤ
52. b6 : Cmd
53. b4 : Id
54. <<<b7, b8>, b6>, b4> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e'@0)
55. f1 = (new_23_sig_when_quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <n, 0> loc(e'@0) <<<b7, b8>, b6>, b4>) ∈ ((Cmd\000C List × (Id List)) ─→ bag(Interface))
56. b1
= rev(mapfilter(λe.(snd(fst(msgval(e))));
λe1.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e1;n;0);
[e;e'@0)))
∈ (Cmd List)
57. b2
= rev(mapfilter(λe.(snd(msgval(e)));λe1.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e1;n;0);[e;e'@0)))
∈ (Id List)
58. ↑(new_23_sig_newvote(Cmd) <n, 0> <<<b7, b8>, b6>, b4> <b1, b2>)
59. ||filter(λe1.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e1;n;0);[e;e'@0))|| = (coeff * flrs) ∈ ℤ
60. w ↓∈ let k,x = poss-maj(eq;[b6 / b1];b6)
in if (k =z (coeff * flrs) + 1)
then new_23_sig_decided'broadcast(Cmd;notify;propose;f) reps <b7, x>
else {new_23_sig_retry'send(Cmd;notify;propose;f) loc(e'@0) <<b7, b8 + 1>, x>}
fi
61. ((coeff * flrs) + 1)
= (||filter(λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);[e;e'))|| + 1)
∈ ℤ
62. e ≤loc e'
63. ↑new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0)
64. ||filter(λe1.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e1;n;0);[e;e'@0))||
= (||filter(λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);[e;e'@0))||
+ ||filter(λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);[e'@0;e'))||)
∈ ℤ
65. (e'@0 ∈ [e, e'])
66. [e;e') = ([e;e'@0) @ [e'@0;e')) ∈ (E List)
67. ↑null(filter(λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);[e'@0;e')))
68. (e'@0 ∈ [e'@0;e'))
69. ↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e'@0;b7;b8;b4)
70. n = b7 ∈ ℤ
71. 0 = b8 ∈ ℤ
72. ↑new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e'@0;b7;b8)
73. ↑new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e'@0;n;0)
74. has-es-info-type(es;e'@0;f;ℤ × ℤ × Cmd × Id)
75. e1 : E@i
76. (e1 ∈ [e;e'@0))@i
77. msgval(e'@0) ∈ ℤ × ℤ × Cmd × Id
78. ↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e1;n;0;snd(msgval(e'@0)))@i
79. e ≤loc e1
80. (e1 <loc e'@0)
81. ↑new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e1;n;0)
82. e ≤loc e1
83. (e1 <loc e'@0)
84. ↑new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e1;n;0)
85. (snd(msgval(e1))) = (snd(msgval(e'@0))) ∈ Id
⊢ b4 = (snd(msgval(e1))) ∈ Id
BY
{ (FLemma `new_23_sig_vote_with_ballot_and_id-snd` [69] THEN Auto) }
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. msgs-interface-delivered-with-omissions(f;es;new\_23\_sig\_main();faulty;flrs;reps)@i
23. bag-no-repeats(Id;reps)@i
24. \#(reps) = ((coeff * flrs) + flrs + 1)@i
25. loc(e) \mdownarrow{}\mmember{} reps@i
26. \mneg{}loc(e) \mdownarrow{}\mmember{} faulty@i
27. <n, c> \mmember{} new\_23\_sig\_Proposal(Cmd;notify;propose;f)(e)@i
28. \mneg{}\muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
29. e' : E
30. ((coeff * flrs) + 1)
= ||mapfilter(\mlambda{}e.<snd(msgval(e)), e, snd(fst(msgval(e)))>
\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e'])||
31. e \mleq{}loc e'
32. \muparrow{}new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0)
33. \mneg{}\muparrow{}(\mexists{}x\mmember{}[e, e'].new\_23\_sig\_decided\_with\_id(Cmd;notify;propose;f;es;x;n))\_b
34. 0 \mleq{} (coeff * flrs)
35. e \mleq{}loc e'
36. \mneg{}\muparrow{}null(filter(\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);[e, e']))
37. \muparrow{}new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e';n;0)
38. has-es-info-type(es;e';f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
39. poss-maj(eq;rev(mapfilter(\mlambda{}e.(snd(fst(msgval(e))));
...;
[e, e']));snd(fst(msgval(e')))) \mmember{} \mBbbZ{} \mtimes{} Cmd
40. e \mleq{}loc e'
41. (fst(poss-maj(eq;rev(mapfilter(\mlambda{}e.(snd(fst(msgval(e))));
\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;...;...;f;es.e;e';n;0);
[e, e']));snd(fst(msgval(e'))))))
= ((coeff * flrs) + 1)
42. loc(e') \mdownarrow{}\mmember{} reps
43. e'@0 : E@i
44. (e'@0 <loc e')@i
45. e \mleq{}loc e'@0 @i
46. w : Interface@i
47. f1 : (Cmd List \mtimes{} (Id List)) {}\mrightarrow{} bag(Interface)
48. b1 : Cmd List
49. b2 : Id List
50. b7 : \mBbbZ{}
51. b8 : \mBbbZ{}
52. b6 : Cmd
53. b4 : Id
54. <<<b7, b8>, b6>, b4> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(e'@0)
55. f1 = (new\_23\_sig\_when\_quorum(Cmd;eq;coeff;flrs;notify;propose;reps;f) <n, 0> loc(e'@0) <<<b7, b8\000C>, b6>, b4>)
56. b1
= rev(mapfilter(\mlambda{}e.(snd(fst(msgval(e))));
\mlambda{}e1.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e1;n;0);
[e;e'@0)))
57. b2
= rev(mapfilter(\mlambda{}e.(snd(msgval(e)));
\mlambda{}e1.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e1;n;0);
[e;e'@0)))
58. \muparrow{}(new\_23\_sig\_newvote(Cmd) <n, 0> <<<b7, b8>, b6>, b4> <b1, b2>)
59. ||filter(\mlambda{}e1.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e1;n;0);[e;e'@0))||
= (coeff * flrs)
60. w \mdownarrow{}\mmember{} let k,x = poss-maj(eq;[b6 / b1];b6)
in if (k =\msubz{} (coeff * flrs) + 1)
then new\_23\_sig\_decided'broadcast(Cmd;notify;propose;f) reps <b7, x>
else \{new\_23\_sig\_retry'send(Cmd;notify;propose;f) loc(e'@0) <<b7, b8 + 1>, x>\}
fi
61. ((coeff * flrs) + 1)
= (||filter(\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);[e;e'))|| + 1)
62. e \mleq{}loc e'
63. \muparrow{}new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0)
64. ||filter(\mlambda{}e1.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e1;n;0);[e;e'@0))||
= (||filter(\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);[e;e'@0))||
+ ||filter(\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);[e'@0;e'))||)
65. (e'@0 \mmember{} [e, e'])
66. [e;e') = ([e;e'@0) @ [e'@0;e'))
67. \muparrow{}null(filter(\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);[e'@0;e')))
68. (e'@0 \mmember{} [e'@0;e'))
69. \muparrow{}new\_23\_sig\_vote\_with\_ballot\_and\_id(Cmd;notify;propose;f;es;e'@0;b7;b8;b4)
70. n = b7
71. 0 = b8
72. \muparrow{}new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e'@0;b7;b8)
73. \muparrow{}new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e'@0;n;0)
74. has-es-info-type(es;e'@0;f;\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
75. e1 : E@i
76. (e1 \mmember{} [e;e'@0))@i
77. msgval(e'@0) \mmember{} \mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id
78. \muparrow{}new\_23\_sig\_vote\_with\_ballot\_and\_id(Cmd;notify;propose;f;es;e1;n;0;snd(msgval(e'@0)))@i
79. e \mleq{}loc e1
80. (e1 <loc e'@0)
81. \muparrow{}new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e1;n;0)
82. e \mleq{}loc e1
83. (e1 <loc e'@0)
84. \muparrow{}new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e1;n;0)
85. (snd(msgval(e1))) = (snd(msgval(e'@0)))
\mvdash{} b4 = (snd(msgval(e1)))
By
Latex:
(FLemma `new\_23\_sig\_vote\_with\_ballot\_and\_id-snd` [69] THEN Auto)
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