Step
*
1
2
2
2
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. e1 : E
35. c' : Cmd
36. e ≤loc e1
37. mk-msg-interface(loc(e);make-Msg(``new_23_sig retry``;<<n, 1>, c'>)) ∈ new_23_sig_main()(e1)
⊢ ↓∃e':E
∃c':Cmd
((e <loc e')
∧ (<n, c'> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e')
∨ (∃r:ℕ+. <<n, r>, c'> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e'))))
BY
{ (UnfoldTopAb 22
THEN RepD
THEN (InstHyp [⌜e1⌝;⌜mk-msg-interface(loc(e);make-Msg(``new_23_sig retry``;<<n, 1>, c'>))⌝] 23⋅ THENA Auto)
THEN SquashExRepD
THEN All(RepUR ``mk-msg-interface``)
THEN OnSomeHyp(\i.RWO "make-msg-interface-equal" i)⋅
THEN Auto
THEN D 0
THEN InstConcl [⌜e2⌝;⌜c'⌝]⋅
THEN Auto
THEN (OrRight THENA 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. e' : E
31. ((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
32. e ≤loc e'
33. ↑new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0)
34. ¬↑(∃x∈[e, e'].new_23_sig_decided_with_id(Cmd;notify;propose;f;es;x;n))_b
35. e1 : E
36. c' : Cmd
37. e ≤loc e1
38. make-msg-interface(0;loc(e);make-Msg(``new_23_sig retry``;<<n, 1>, c'>)) ∈ new_23_sig_main()(e1)
39. e2 : E
40. delay : ℤ
41. (e1 < e2)
42. 0 = delay ∈ ℤ
43. loc(e) = loc(e2) ∈ Id
44. make-Msg(``new_23_sig retry``;<<n, 1>, c'>) = info(e2) ∈ Message(f)
45. (e <loc e2)
⊢ ∃r:ℕ+. <<n, r>, c'> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e2)
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. e1 : E
35. c' : Cmd
36. e \mleq{}loc e1
37. mk-msg-interface(loc(e);make-Msg(``new\_23\_sig retry``;<<n, 1>, c'>)) \mmember{} new\_23\_sig\_main()(e1)
\mvdash{} \mdownarrow{}\mexists{}e':E
\mexists{}c':Cmd
((e <loc e')
\mwedge{} (<n, c'> \mmember{} new\_23\_sig\_decided'base(Cmd;notify;propose;f)(e')
\mvee{} (\mexists{}r:\mBbbN{}\msupplus{}. <<n, r>, c'> \mmember{} new\_23\_sig\_retry'base(Cmd;notify;propose;f)(e'))))
By
Latex:
(UnfoldTopAb 22
THEN RepD
THEN (InstHyp [\mkleeneopen{}e1\mkleeneclose{};\mkleeneopen{}mk-msg-interface(loc(e);make-Msg(``new\_23\_sig retry``;<<n, 1>, c'>))\mkleeneclose{}] 23\mcdot{} THE\000CNA Auto)
THEN SquashExRepD
THEN All(RepUR ``mk-msg-interface``)
THEN OnSomeHyp(\mbackslash{}i.RWO "make-msg-interface-equal" i)\mcdot{}
THEN Auto
THEN D 0
THEN InstConcl [\mkleeneopen{}e2\mkleeneclose{};\mkleeneopen{}c'\mkleeneclose{}]\mcdot{}
THEN Auto
THEN (OrRight THENA Auto))
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