Step
*
1
1
of Lemma
new_23_sig_progress-step9-v2
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. b_all(Id;[x∈reps|
¬bbag-deq-member(IdDeq;x;faulty)];i.(i
∈ mapfilter(λe.(snd(msgval(e)));
λe.new_23_sig_vote_with_ballot(Cmd;...;...;f;es;e;n;0);
[e, e'])))
31. ((coeff * flrs) + 1) ≤ ||mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
λe.new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e;n;0);
[e, e'])||
⊢ ↓∃e':E
(((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)
BY
{ ((Assert ⌈0 ≤ (coeff * flrs)⌉⋅ THENA Auto)
THEN (Assert ⌈((coeff * flrs) + 1) ≤ #([x∈reps|¬bbag-deq-member(IdDeq;x;faulty)])⌉⋅
THENA ((Assert ⌈#(reps)
= (#([x∈reps|bag-deq-member(IdDeq;x;faulty)]) + #([x∈reps|¬bbag-deq-member(IdDeq;x;faulty)]))
∈ ℤ⌉⋅
THENA ((RWO "bag-size-append<" 0 THENA Auto)
THEN (InstLemma `bag-filter-split` [⌈Id⌉;⌈λ2x.bag-deq-member(IdDeq;x;faulty)⌉;⌈reps⌉]⋅
THENA Auto
)
THEN HypSubst (-1) 0
THEN Auto)
)
THEN D 22
THEN InstLemma `bag-size-filter-member-bound` [⌈Id⌉;⌈IdDeq⌉;⌈reps⌉;⌈faulty⌉]⋅
THEN Auto)
)
THEN (FLemma `sub-bag-list-if-bag-no-repeats-sq` [-4]
THENA (Auto
THEN Try ((BLemma `bag-filter-no-repeats` THEN Auto))
THEN Reduce (-1)
THEN FLemma `new_23_sig_vote_with_ballot-assert-type` [-1]
THEN Auto)
)
THEN D (-1)
THEN (InstLemma `sub-bag-remove-repeats-if-no-repeats` [⌈Id⌉;⌈IdDeq⌉;⌈[x∈reps|¬bbag-deq-member(IdDeq;x;faulty)]⌉;
⌈mapfilter(λe.(snd(msgval(e)));λe.new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e;n;0);[e, e'])⌉]⋅
THENA (Auto
THEN Try ((BLemma `bag-no-repeats-filter` THEN Auto))
THEN Assert ⌈mapfilter(λe.(snd(msgval(e)));
λe.new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e;n;0);
[e, e']) ∈ Id List⌉⋅
THEN Auto
THEN Reduce (-1)
THEN FLemma `new_23_sig_vote_with_ballot-assert-type` [-1]
THEN Auto)
)
THEN Unfold `bag-remove-repeats` (-1)) }
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. b_all(Id;[x∈reps|
¬bbag-deq-member(IdDeq;x;faulty)];i.(i
∈ mapfilter(λe.(snd(msgval(e)));
λe.new_23_sig_vote_with_ballot(Cmd;...;...;f;es;e;n;0);
[e, e'])))
31. ((coeff * flrs) + 1) ≤ ||mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
λe.new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e;n;0);
[e, e'])||
32. 0 ≤ (coeff * flrs)
33. ((coeff * flrs) + 1) ≤ #([x∈reps|¬bbag-deq-member(IdDeq;x;faulty)])
34. sub-bag(Id;[x∈reps|
¬bbag-deq-member(IdDeq;x;faulty)];mapfilter(λe.(snd(msgval(e)));
λe.new_23_sig_vote_with_ballot(Cmd;notify;...;f;es;e;n;0);
[e, e']))
35. sub-bag(Id;[x∈reps|
¬bbag-deq-member(IdDeq;x;faulty)];list-to-set(IdDeq;mapfilter(λe.(snd(msgval(e)));
λe....;
[e, e'])))
⊢ ↓∃e':E
(((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)
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. b\_all(Id;[x\mmember{}reps|
\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)];i.(i
\mmember{} mapfilter(\mlambda{}e.(snd(msgval(e)));
\mlambda{}e....;
[e, e'])))
31. ((coeff * flrs) + 1)
\mleq{} ||mapfilter(\mlambda{}e.<snd(msgval(e)), e, snd(fst(msgval(e)))>
\mlambda{}e.new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e;n;0);
[e, e'])||
\mvdash{} \mdownarrow{}\mexists{}e':E
(((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'])||)
By
Latex:
((Assert \mkleeneopen{}0 \mleq{} (coeff * flrs)\mkleeneclose{}\mcdot{} THENA Auto)
THEN (Assert \mkleeneopen{}((coeff * flrs) + 1) \mleq{} \#([x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)])\mkleeneclose{}\mcdot{}
THENA ((Assert \mkleeneopen{}\#(reps)
= (\#([x\mmember{}reps|bag-deq-member(IdDeq;x;faulty)])
+ \#([x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)]))\mkleeneclose{}\mcdot{}
THENA ((RWO "bag-size-append<" 0 THENA Auto)
THEN (InstLemma `bag-filter-split` [\mkleeneopen{}Id\mkleeneclose{};\mkleeneopen{}\mlambda{}\msubtwo{}x.bag-deq-member(IdDeq;x;faulty)\mkleeneclose{};
\mkleeneopen{}reps\mkleeneclose{}]\mcdot{}
THENA Auto
)
THEN HypSubst (-1) 0
THEN Auto)
)
THEN D 22
THEN InstLemma `bag-size-filter-member-bound` [\mkleeneopen{}Id\mkleeneclose{};\mkleeneopen{}IdDeq\mkleeneclose{};\mkleeneopen{}reps\mkleeneclose{};\mkleeneopen{}faulty\mkleeneclose{}]\mcdot{}
THEN Auto)
)
THEN (FLemma `sub-bag-list-if-bag-no-repeats-sq` [-4]
THENA (Auto
THEN Try ((BLemma `bag-filter-no-repeats` THEN Auto))
THEN Reduce (-1)
THEN FLemma `new\_23\_sig\_vote\_with\_ballot-assert-type` [-1]
THEN Auto)
)
THEN D (-1)
THEN (InstLemma `sub-bag-remove-repeats-if-no-repeats` [\mkleeneopen{}Id\mkleeneclose{};\mkleeneopen{}IdDeq\mkleeneclose{};
\mkleeneopen{}[x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)]\mkleeneclose{};
\mkleeneopen{}mapfilter(\mlambda{}e.(snd(msgval(e)));
\mlambda{}e.new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e;n;0);
[e, e'])\mkleeneclose{}]\mcdot{}
THENA (Auto
THEN Try ((BLemma `bag-no-repeats-filter` THEN Auto))
THEN Assert \mkleeneopen{}mapfilter(\mlambda{}e.(snd(msgval(e)));
\mlambda{}e.new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e;n;0);
[e, e']) \mmember{} Id List\mkleeneclose{}\mcdot{}
THEN Auto
THEN Reduce (-1)
THEN FLemma `new\_23\_sig\_vote\_with\_ballot-assert-type` [-1]
THEN Auto)
)
THEN Unfold `bag-remove-repeats` (-1))
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