Step
*
1
1
of Lemma
new_23_sig_progress-step5
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. b_all(Id;[x∈reps|¬bbag-deq-member(IdDeq;x;faulty)];i.↓∃e':E
∃c':Cmd
((loc(e') = loc(e) ∈ Id)
∧ <<<n, 0>, c'>, i> ∈
new_23_sig_vote'base(Cmd;notify;propose;f)(e')))
⊢ ↓∃bs:(Id × E × Cmd) List
((bag-map(λi.(fst(i));bs) = [x∈reps|¬bbag-deq-member(IdDeq;x;faulty)] ∈ bag(Id))
∧ (∀x∈bs.(loc(fst(snd(x))) = loc(e) ∈ Id)
∧ <<<n, 0>, snd(snd(x))>, fst(x)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(fst(snd(x)))))
BY
{ ((FLemma `b_all-squash-exists-list` [-1] THENA Auto)
THEN SquashExRepD
THEN (FLemma `l_all-squash-exists-list` [-1] THENA Auto)
THEN SquashExRepD
THEN D 0
THEN InstConcl [⌜map(λx.let z,c = x
in let i,e = z
in <i, e, c>;b1)⌝]⋅
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. b_all(Id;[x∈reps|¬bbag-deq-member(IdDeq;x;faulty)];i.↓∃e':E
∃c':Cmd
((loc(e') = loc(e) ∈ Id)
∧ <<<n, 0>, c'>, i> ∈
new_23_sig_vote'base(Cmd;notify;propose;f)(e')))
30. bs : (Id × E) List
31. bag-map(λi.(fst(i));bs) = [x∈reps|¬bbag-deq-member(IdDeq;x;faulty)] ∈ bag(Id)
32. (∀i∈bs.↓∃c':Cmd
((loc(snd(i)) = loc(e) ∈ Id)
∧ <<<n, 0>, c'>, fst(i)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(snd(i))))
33. b1 : (Id × E × Cmd) List
34. map(λi.(fst(i));b1) = bs ∈ ((Id × E) List)
35. (∀i∈b1.↓(loc(snd(fst(i))) = loc(e) ∈ Id)
∧ <<<n, 0>, snd(i)>, fst(fst(i))> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(snd(fst(i))))
⊢ bag-map(λi.(fst(i));map(λx.let z,c = x
in let i,e = z
in <i, e, c>;b1))
= [x∈reps|¬bbag-deq-member(IdDeq;x;faulty)]
∈ bag(Id)
2
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. b_all(Id;[x∈reps|¬bbag-deq-member(IdDeq;x;faulty)];i.↓∃e':E
∃c':Cmd
((loc(e') = loc(e) ∈ Id)
∧ <<<n, 0>, c'>, i> ∈
new_23_sig_vote'base(Cmd;notify;propose;f)(e')))
30. bs : (Id × E) List
31. bag-map(λi.(fst(i));bs) = [x∈reps|¬bbag-deq-member(IdDeq;x;faulty)] ∈ bag(Id)
32. (∀i∈bs.↓∃c':Cmd
((loc(snd(i)) = loc(e) ∈ Id)
∧ <<<n, 0>, c'>, fst(i)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(snd(i))))
33. b1 : (Id × E × Cmd) List
34. map(λi.(fst(i));b1) = bs ∈ ((Id × E) List)
35. (∀i∈b1.↓(loc(snd(fst(i))) = loc(e) ∈ Id)
∧ <<<n, 0>, snd(i)>, fst(fst(i))> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(snd(fst(i))))
36. bag-map(λi.(fst(i));map(λx.let z,c = x
in let i,e = z
in <i, e, c>;b1))
= [x∈reps|¬bbag-deq-member(IdDeq;x;faulty)]
∈ bag(Id)
⊢ (∀x∈map(λx.let z,c = x
in let i,e = z
in <i, e, c>;b1).(loc(fst(snd(x))) = loc(e) ∈ Id)
∧ <<<n, 0>, snd(snd(x))>, fst(x)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(fst(snd(x))))
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. b\_all(Id;[x\mmember{}reps|
\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)];i.\mdownarrow{}\mexists{}e':E
\mexists{}c':Cmd
((loc(e') = loc(e))
\mwedge{} <<<n, 0>, c'>, i> \mmember{}
new\_23\_sig\_vote'base(Cmd;notify;propose;f)(
e')))
\mvdash{} \mdownarrow{}\mexists{}bs:(Id \mtimes{} E \mtimes{} Cmd) List
((bag-map(\mlambda{}i.(fst(i));bs) = [x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)])
\mwedge{} (\mforall{}x\mmember{}bs.(loc(fst(snd(x))) = loc(e))
\mwedge{} <<<n, 0>, snd(snd(x))>, fst(x)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(
fst(snd(x)))))
By
Latex:
((FLemma `b\_all-squash-exists-list` [-1] THENA Auto)
THEN SquashExRepD
THEN (FLemma `l\_all-squash-exists-list` [-1] THENA Auto)
THEN SquashExRepD
THEN D 0
THEN InstConcl [\mkleeneopen{}map(\mlambda{}x.let z,c = x
in let i,e = z
in <i, e, c>b1)\mkleeneclose{}]\mcdot{}
THEN Auto)
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