Step
*
1
1
2
of Lemma
new_23_sig_progress-step9
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. bs : (Id × E × Cmd) List
30. ((coeff * flrs) + 1) ≤ ||bs||
31. (∀x∈bs.(loc(fst(snd(x))) = loc(e) ∈ Id)
∧ e ≤loc fst(snd(x))
∧ <<<n, 0>, snd(snd(x))>, fst(x)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(fst(snd(x))))
32. l-ordered(Id × E × Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
33. (∀e'∈[e, fst(snd(last(bs)))].(↑new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e';n;0))
⇒ (e' ∈ map(λx.(fst(snd(x)));bs)))
34. b_all(Id;[x∈reps|¬bbag-deq-member(IdDeq;x;faulty)];i.(i ∈ map(λi.(fst(i));bs)))
35. 0 ≤ (coeff * flrs)
36. ((coeff * flrs) + 1) ≤ ||remove-repeats-fun(IdDeq;λx.(fst(x));bs)||
⊢ ↓∃bs:(Id × E × Cmd) List
((((coeff * flrs) + 1) = ||bs|| ∈ ℤ)
∧ (∀x∈bs.(loc(fst(snd(x))) = loc(e) ∈ Id)
∧ e ≤loc fst(snd(x))
∧ <<<n, 0>, snd(snd(x))>, fst(x)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(fst(snd(x)))
∧ (∀e'∈[e;fst(snd(x))).¬↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e';n;0;fst(x))))
∧ l-ordered(Id × E × Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
∧ (∀e'∈[e, fst(snd(last(bs)))].∀i:Id
((↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e';n;0;i))
⇒ (∀e''∈[e;e').
¬↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e'';n;0;i))
⇒ (e' ∈ map(λx.(fst(snd(x)));bs))))
∧ no_repeats(Id;map(λx.(fst(x));bs)))
BY
{ ((InstLemma `list-decomp-nat` [⌈Id × E × Cmd⌉;⌈remove-repeats-fun(IdDeq;λx.(fst(x));bs)⌉;⌈(coeff * flrs) + 1⌉]⋅
THENA Auto
)
THEN ExRepD
THEN (Assert ⌈¬↑null(K)⌉⋅ THENA (BLemma `non_null_iff_length` THEN Auto))
THEN (Assert ⌈remove-repeats-fun(IdDeq;λx.(fst(x));bs) ⊆ bs⌉⋅ THENA (BLemma `remove-repeats-fun-sublist` THEN Auto))
THEN (Assert ⌈K ⊆ remove-repeats-fun(IdDeq;λx.(fst(x));bs)⌉⋅ THENA (HypSubst (-4) 0 THEN Auto))
THEN (Assert ⌈K ⊆ bs⌉⋅ THENA (FLemma `sublist_transitivity` [-1;-2] 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. bs : (Id × E × Cmd) List
30. ((coeff * flrs) + 1) ≤ ||bs||
31. (∀x∈bs.(loc(fst(snd(x))) = loc(e) ∈ Id)
∧ e ≤loc fst(snd(x))
∧ <<<n, 0>, snd(snd(x))>, fst(x)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(fst(snd(x))))
32. l-ordered(Id × E × Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
33. (∀e'∈[e, fst(snd(last(bs)))].(↑new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e';n;0))
⇒ (e' ∈ map(λx.(fst(snd(x)));bs)))
34. b_all(Id;[x∈reps|¬bbag-deq-member(IdDeq;x;faulty)];i.(i ∈ map(λi.(fst(i));bs)))
35. 0 ≤ (coeff * flrs)
36. ((coeff * flrs) + 1) ≤ ||remove-repeats-fun(IdDeq;λx.(fst(x));bs)||
37. K : (Id × E × Cmd) List
38. J : (Id × E × Cmd) List
39. remove-repeats-fun(IdDeq;λx.(fst(x));bs) = (K @ J) ∈ ((Id × E × Cmd) List)
40. ||K|| = ((coeff * flrs) + 1) ∈ ℤ
41. ¬↑null(K)
42. remove-repeats-fun(IdDeq;λx.(fst(x));bs) ⊆ bs
43. K ⊆ remove-repeats-fun(IdDeq;λx.(fst(x));bs)
44. K ⊆ bs
⊢ ↓∃bs:(Id × E × Cmd) List
((((coeff * flrs) + 1) = ||bs|| ∈ ℤ)
∧ (∀x∈bs.(loc(fst(snd(x))) = loc(e) ∈ Id)
∧ e ≤loc fst(snd(x))
∧ <<<n, 0>, snd(snd(x))>, fst(x)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(fst(snd(x)))
∧ (∀e'∈[e;fst(snd(x))).¬↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e';n;0;fst(x))))
∧ l-ordered(Id × E × Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
∧ (∀e'∈[e, fst(snd(last(bs)))].∀i:Id
((↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e';n;0;i))
⇒ (∀e''∈[e;e').
¬↑new_23_sig_vote_with_ballot_and_id(Cmd;notify;propose;f;es;e'';n;0;i))
⇒ (e' ∈ map(λx.(fst(snd(x)));bs))))
∧ no_repeats(Id;map(λx.(fst(x));bs)))
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. bs : (Id \mtimes{} E \mtimes{} Cmd) List
30. ((coeff * flrs) + 1) \mleq{} ||bs||
31. (\mforall{}x\mmember{}bs.(loc(fst(snd(x))) = loc(e))
\mwedge{} e \mleq{}loc fst(snd(x))
\mwedge{} <<<n, 0>, snd(snd(x))>, fst(x)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(fst(snd(x))))
32. l-ordered(Id \mtimes{} E \mtimes{} Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
33. (\mforall{}e'\mmember{}[e, fst(snd(last(bs)))].(\muparrow{}new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e';n;0))
{}\mRightarrow{} (e' \mmember{} map(\mlambda{}x.(fst(snd(x)));bs)))
34. b\_all(Id;[x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)];i.(i \mmember{} map(\mlambda{}i.(fst(i));bs)))
35. 0 \mleq{} (coeff * flrs)
36. ((coeff * flrs) + 1) \mleq{} ||remove-repeats-fun(IdDeq;\mlambda{}x.(fst(x));bs)||
\mvdash{} \mdownarrow{}\mexists{}bs:(Id \mtimes{} E \mtimes{} Cmd) List
((((coeff * flrs) + 1) = ||bs||)
\mwedge{} (\mforall{}x\mmember{}bs.(loc(fst(snd(x))) = loc(e))
\mwedge{} e \mleq{}loc fst(snd(x))
\mwedge{} <<<n, 0>, snd(snd(x))>, fst(x)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(fst(snd(x)))
\mwedge{} (\mforall{}e'\mmember{}[e;fst(snd(x))).
\mneg{}\muparrow{}new\_23\_sig\_vote\_with\_ballot\_and\_id(Cmd;notify;propose;f;es;e';n;0;fst(x))))
\mwedge{} l-ordered(Id \mtimes{} E \mtimes{} Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
\mwedge{} (\mforall{}e'\mmember{}[e, fst(snd(last(bs)))].
\mforall{}i:Id
((\muparrow{}new\_23\_sig\_vote\_with\_ballot\_and\_id(Cmd;notify;propose;f;es;e';n;0;i))
{}\mRightarrow{} (\mforall{}e''\mmember{}[e;e').\mneg{}\muparrow{}new\_23\_sig\_vote\_with\_ballot\_and\_id(Cmd;notify;propose;f;es;e'';n;0;i))
{}\mRightarrow{} (e' \mmember{} map(\mlambda{}x.(fst(snd(x)));bs))))
\mwedge{} no\_repeats(Id;map(\mlambda{}x.(fst(x));bs)))
By
Latex:
((InstLemma `list-decomp-nat` [\mkleeneopen{}Id \mtimes{} E \mtimes{} Cmd\mkleeneclose{};\mkleeneopen{}remove-repeats-fun(IdDeq;\mlambda{}x.(fst(x));bs)\mkleeneclose{};\mkleeneopen{}(coeff
* flrs)
+ 1\mkleeneclose{}]\mcdot{}
THENA Auto
)
THEN ExRepD
THEN (Assert \mkleeneopen{}\mneg{}\muparrow{}null(K)\mkleeneclose{}\mcdot{} THENA (BLemma `non\_null\_iff\_length` THEN Auto))
THEN (Assert \mkleeneopen{}remove-repeats-fun(IdDeq;\mlambda{}x.(fst(x));bs) \msubseteq{} bs\mkleeneclose{}\mcdot{}
THENA (BLemma `remove-repeats-fun-sublist` THEN Auto)
)
THEN (Assert \mkleeneopen{}K \msubseteq{} remove-repeats-fun(IdDeq;\mlambda{}x.(fst(x));bs)\mkleeneclose{}\mcdot{} THENA (HypSubst (-4) 0 THEN Auto))
THEN (Assert \mkleeneopen{}K \msubseteq{} bs\mkleeneclose{}\mcdot{} THENA (FLemma `sublist\_transitivity` [-1;-2] THEN Auto)))\mcdot{}
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