Proof of Nuprl Lemma : new_23_sig

Step * 1 1 2 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')))

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))))
BY
{ (RWO "l_all_iff" 0
   THEN Auto
   THEN DProdsAndUnions
   THEN AllReduce
   THEN OnSomeHyp(\i.(RW ListC i THENA Auto))⋅
   THEN ExRepD
   THEN DProdsAndUnions
   THEN AllReduce
   THEN All(\i.(RWO "l_all_iff" i THENA Auto))⋅
   THEN OnSomeHyp(\i.(InstHyp [⌜<<y3, y4>, y2>⌝] i⋅ THENA CpltAuto))⋅
   THEN TrySquashExRepD (-1)
   THEN AllReduce
   THEN Auto) }

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'))) 30. bs : (Id \mtimes{} E) List 31. bag-map(\mlambda{}i.(fst(i));bs) = [x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)] 32. (\mforall{}i\mmember{}bs.\mdownarrow{}\mexists{}c':Cmd ((loc(snd(i)) = loc(e)) \mwedge{} <<<n, 0>, c'>, fst(i)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(snd(i)))) 33. b1 : (Id \mtimes{} E \mtimes{} Cmd) List 34. map(\mlambda{}i.(fst(i));b1) = bs 35. (\mforall{}i\mmember{}b1.\mdownarrow{}(loc(snd(fst(i))) = loc(e)) \mwedge{} <<<n, 0>, snd(i)>, fst(fst(i))> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)( snd(fst(i)))) 36. bag-map(\mlambda{}i.(fst(i));map(\mlambda{}x.let z,c = x in let i,e = z in <i, e, c>b1)) = [x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)] \mvdash{} (\mforall{}x\mmember{}map(\mlambda{}x.let z,c = x in let i,e = z in <i, e, c>b1).(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: (RWO "l\_all\_iff" 0 THEN Auto THEN DProdsAndUnions THEN AllReduce THEN OnSomeHyp(\mbackslash{}i.(RW ListC i THENA Auto))\mcdot{} THEN ExRepD THEN DProdsAndUnions THEN AllReduce THEN All(\mbackslash{}i.(RWO "l\_all\_iff" i THENA Auto))\mcdot{} THEN OnSomeHyp(\mbackslash{}i.(InstHyp [\mkleeneopen{}<<y3, y4>, y2>\mkleeneclose{}] i\mcdot{} THENA CpltAuto))\mcdot{} THEN TrySquashExRepD (-1) THEN AllReduce THEN Auto) Home Index