Step
*
1
2
1
of Lemma
new_23_sig_progress-step10
.....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. (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
⊢ ∃e':E
∃c':Cmd
(e ≤loc e'
∧ (mk-msg-interface(loc(e);make-Msg(``new_23_sig decided``;<n, c'>)) ∈ new_23_sig_main()(e') ∨ mk-msg-interface(loc(\000Ce);make-Msg(``new_23_sig retry``;<<n, 1>, c'>)) ∈ new_23_sig_main()(e')))
BY
{ ((Assert ⌜0 ≤ (coeff * flrs)⌝⋅ THENA Auto)
THEN (Assert ⌜e ≤loc e' ⌝⋅
THENA ((Decide ⌜loc(e) = loc(e') ∈ Id⌝⋅ THENA Auto)
THEN Try (((FLemma `es-interval-diff-loc` [-1] THENA Auto)
THEN HypSubst (-1) (-5)
THEN Reduce (-5)
THEN Auto))
THEN SupposeNot
THEN (Assert ⌜False⌝⋅ THEN Auto)
THEN (RWO "es-locl-not-le<" (-1) THENA Auto)
THEN (FLemma `es-interval-is-nil` [-1] THENA Auto)
THEN HypSubst (-1) (-6)
THEN Reduce (-6)
THEN Auto)
)
THEN (Assert ⌜¬↑null(filter(λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);[e, e']))⌝⋅
THENA ((BLemma `non_null_iff_length` THEN Auto)
THEN Try (Complete ((RW ListC (-2) THEN Auto)))
THEN Unfold `mapfilter` (-6)
THEN (RWO "length-map" (-6) THENA Auto)
THEN Try ((RevHypSubst (-6) 0 THEN Auto))
THEN RW ListC (-2)
THEN Auto)
)
THEN (FLemma `new_23_sig_vote_with_ballot_first-assert-forward` [-5] THENA Auto)
THEN (FLemma `new_23_sig_vote_with_ballot-assert-type` [-1] THENA Auto)
THEN (Assert ⌜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⌝⋅
THENA (RepeatFor 2 (Try (MemCD))
THEN Try (BLemma `mapfilter-wf2`)
THEN AllReduce
THEN Auto
THEN Try (Complete ((RW ListC (-2) THEN Auto)))
THEN DVarSets
THEN (RWO "new_23_sig_vote_with_ballot_first-forward" (-1) THEN Auto)
THEN Try (Complete ((RW ListC (-2) THEN Auto)))
THEN FLemma `new_23_sig_vote_with_ballot-assert-type` [-1]
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. 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
⊢ ∃e':E
∃c':Cmd
(e ≤loc e'
∧ (mk-msg-interface(loc(e);make-Msg(``new_23_sig decided``;<n, c'>)) ∈ new_23_sig_main()(e') ∨ mk-msg-interface(loc(\000Ce);make-Msg(``new_23_sig retry``;<<n, 1>, c'>)) ∈ new_23_sig_main()(e')))
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. (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
\mvdash{} \mexists{}e':E
\mexists{}c':Cmd
(e \mleq{}loc e'
\mwedge{} (mk-msg-interface(loc(e);make-Msg(``new\_23\_sig decided``;<n, c'>)) \mmember{} new\_23\_sig\_main()(e')
\mvee{} mk-msg-interface(loc(e);make-Msg(``new\_23\_sig retry``;<<n, 1>, c'>)) \mmember{} new\_23\_sig\_main()(e')\000C))
By
Latex:
((Assert \mkleeneopen{}0 \mleq{} (coeff * flrs)\mkleeneclose{}\mcdot{} THENA Auto)
THEN (Assert \mkleeneopen{}e \mleq{}loc e' \mkleeneclose{}\mcdot{}
THENA ((Decide \mkleeneopen{}loc(e) = loc(e')\mkleeneclose{}\mcdot{} THENA Auto)
THEN Try (((FLemma `es-interval-diff-loc` [-1] THENA Auto)
THEN HypSubst (-1) (-5)
THEN Reduce (-5)
THEN Auto))
THEN SupposeNot
THEN (Assert \mkleeneopen{}False\mkleeneclose{}\mcdot{} THEN Auto)
THEN (RWO "es-locl-not-le<" (-1) THENA Auto)
THEN (FLemma `es-interval-is-nil` [-1] THENA Auto)
THEN HypSubst (-1) (-6)
THEN Reduce (-6)
THEN Auto)
)
THEN (Assert
\mkleeneopen{}\mneg{}\muparrow{}null(filter(\mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e']))\mkleeneclose{}\mcdot{}
THENA ((BLemma `non\_null\_iff\_length` THEN Auto)
THEN Try (Complete ((RW ListC (-2) THEN Auto)))
THEN Unfold `mapfilter` (-6)
THEN (RWO "length-map" (-6) THENA Auto)
THEN Try ((RevHypSubst (-6) 0 THEN Auto))
THEN RW ListC (-2)
THEN Auto)
)
THEN (FLemma `new\_23\_sig\_vote\_with\_ballot\_first-assert-forward` [-5] THENA Auto)
THEN (FLemma `new\_23\_sig\_vote\_with\_ballot-assert-type` [-1] THENA Auto)
THEN (Assert
\mkleeneopen{}poss-maj(eq;rev(mapfilter(\mlambda{}e.(snd(fst(msgval(e))));
...;
[e, e']));snd(fst(msgval(e')))) \mmember{} \mBbbZ{} \mtimes{} Cmd\mkleeneclose{}\mcdot{}
THENA (RepeatFor 2 (Try (MemCD))
THEN Try (BLemma `mapfilter-wf2`)
THEN AllReduce
THEN Auto
THEN Try (Complete ((RW ListC (-2) THEN Auto)))
THEN DVarSets
THEN (RWO "new\_23\_sig\_vote\_with\_ballot\_first-forward" (-1) THEN Auto)
THEN Try (Complete ((RW ListC (-2) THEN Auto)))
THEN FLemma `new\_23\_sig\_vote\_with\_ballot-assert-type` [-1]
THEN Auto)
))
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