Proof of Nuprl Lemma : new_23_sig

Step * 1 2 of Lemma new_23_sig_progress-step7

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. bag-map(λi.(fst(i));bs) = [x∈reps|¬_bbag-deq-member(IdDeq;x;faulty)] ∈ bag(Id)
31. (∀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))))
32. l-ordered(Id × E × Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
33. bag-map(λi.(fst(i));bs) = [x∈reps|¬_bbag-deq-member(IdDeq;x;faulty)] ∈ bag(Id)
34. (∀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))))
35. l-ordered(Id × E × Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs)
36. (∀x∈bs.e ≤loc fst(snd(x)) )
⊢ ((coeff * flrs) + 1) ≤ ||bs||
BY
{ ((Subst ⌈||bs|| = ||map(λx.(fst(x));bs)|| ∈ ℤ⌉ 0⋅ THENA (Auto THEN RW ListC 0 THEN Auto))
   THEN Fold `bag-map` 0
   THEN Fold `bag-size` 0

   THEN (Subst ⌈#(bag-map(λx.(fst(x));bs)) = #([x∈reps|¬_bbag-deq-member(IdDeq;x;faulty)]) ∈ ℤ⌉ 0⋅

         THENA (Auto THEN HypSubst (-7) 0 THEN Auto)
         )
   THEN (Assert ⌈#([x∈reps|bag-deq-member(IdDeq;x;faulty)] + [x∈reps|¬_bbag-deq-member(IdDeq;x;faulty)])
                 = ((coeff * flrs) + flrs + 1)
                 ∈ ℤ⌉⋅

         THENA ((InstLemma `bag-filter-split` [⌈Id⌉;⌈λ₂x.bag-deq-member(IdDeq;x;faulty)⌉;⌈reps⌉]⋅ THENA Auto)

                THEN HypSubst (-1) 0
                THEN Auto)
         )
   THEN (RWO "bag-size-append" (-1) THENA Auto)
   THEN (Assert ⌈#([x∈reps|bag-deq-member(IdDeq;x;faulty)]) ≤ flrs⌉⋅ THENM Auto')
   THEN D 22
   THEN (Assert ⌈#([x∈reps|bag-deq-member(IdDeq;x;faulty)]) ≤ #(faulty)⌉⋅ THENM Auto)
   THEN BLemma `bag-size-filter-member-bound`
   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. bs : (Id \mtimes{} E \mtimes{} Cmd) List 30. bag-map(\mlambda{}i.(fst(i));bs) = [x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)] 31. (\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)))) 32. l-ordered(Id \mtimes{} E \mtimes{} Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs) 33. bag-map(\mlambda{}i.(fst(i));bs) = [x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)] 34. (\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)))) 35. l-ordered(Id \mtimes{} E \mtimes{} Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));bs) 36. (\mforall{}x\mmember{}bs.e \mleq{}loc fst(snd(x)) ) \mvdash{} ((coeff * flrs) + 1) \mleq{} ||bs|| By Latex: ((Subst \mkleeneopen{}||bs|| = ||map(\mlambda{}x.(fst(x));bs)||\mkleeneclose{} 0\mcdot{} THENA (Auto THEN RW ListC 0 THEN Auto)) THEN Fold `bag-map` 0 THEN Fold `bag-size` 0 THEN (Subst \mkleeneopen{}\#(bag-map(\mlambda{}x.(fst(x));bs)) = \#([x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)])\mkleeneclose{} 0\mcdot{} THENA (Auto THEN HypSubst (-7) 0 THEN Auto) ) THEN (Assert \mkleeneopen{}\#([x\mmember{}reps|bag-deq-member(IdDeq;x;faulty)] + [x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)]) = ((coeff * flrs) + flrs + 1)\mkleeneclose{}\mcdot{} THENA ((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 (RWO "bag-size-append" (-1) THENA Auto) THEN (Assert \mkleeneopen{}\#([x\mmember{}reps|bag-deq-member(IdDeq;x;faulty)]) \mleq{} flrs\mkleeneclose{}\mcdot{} THENM Auto') THEN D 22 THEN (Assert \mkleeneopen{}\#([x\mmember{}reps|bag-deq-member(IdDeq;x;faulty)]) \mleq{} \#(faulty)\mkleeneclose{}\mcdot{} THENM Auto) THEN BLemma `bag-size-filter-member-bound` THEN Auto) Home Index