Proof of Nuprl Lemma : new_23_sig

Step * 1 1 1 of Lemma new_23_sig_progress-step9

.....assertion.....
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)
⊢ ((coeff * flrs) + 1) ≤ ||remove-repeats-fun(IdDeq;λx.(fst(x));bs)||
BY
{ ((RWO "remove-repeats-fun-length-as-remove-repeats-map" 0 THENA Auto)

   THEN (FLemma `sub-bag-list-if-bag-no-repeats-sq` [-2] THENA (Auto THEN BLemma `bag-filter-no-repeats` THEN Auto))

   THEN D (-1)
   THEN (Unhide THENA Auto)
   THEN Fold `bag-size` 0⋅
   THEN D (-1)
   THEN (RWO "bag-remove-repeats-eq-remove-repeats<" 0 THENA Auto)
   THEN (HypSubst (-1) 0 THENA Auto)
   THEN (RWO "bag-remove-repeats-append" 0 THENA Auto)
   THEN (RWO "bag-size-append" 0 THENA Auto)
   THEN GenConclAtAddr [2;2]

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

THEN (HypSubst (-1) 0 THENA Auto)
THEN (RWO "bag-remove-repeats-if-no-repeats" 0 THENA Auto)

   THEN (Assert ⌈((coeff * flrs) + 1) ≤ #([x∈reps|¬_bbag-deq-member(IdDeq;x;faulty)])⌉⋅ THEN Auto)

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

THEN (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 HypSubst (-1) 0 THEN Auto)
         )
   THEN (Assert ⌈#([x∈reps|bag-deq-member(IdDeq;x;faulty)]) ≤ flrs⌉⋅ THEN Auto)
   THEN D 22
   THEN InstLemma `bag-size-filter-member-bound` [⌈Id⌉;⌈IdDeq⌉;⌈reps⌉;⌈faulty⌉]⋅
   THEN Auto)⋅ }

Latex:

Latex:
.....assertion..... 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) \mvdash{} ((coeff * flrs) + 1) \mleq{} ||remove-repeats-fun(IdDeq;\mlambda{}x.(fst(x));bs)|| By Latex: ((RWO "remove-repeats-fun-length-as-remove-repeats-map" 0 THENA Auto) THEN (FLemma `sub-bag-list-if-bag-no-repeats-sq` [-2] THENA (Auto THEN BLemma `bag-filter-no-repeats` THEN Auto) ) THEN D (-1) THEN (Unhide THENA Auto) THEN Fold `bag-size` 0\mcdot{} THEN D (-1) THEN (RWO "bag-remove-repeats-eq-remove-repeats<" 0 THENA Auto) THEN (HypSubst (-1) 0 THENA Auto) THEN (RWO "bag-remove-repeats-append" 0 THENA Auto) THEN (RWO "bag-size-append" 0 THENA Auto) THEN GenConclAtAddr [2;2] THEN (InstLemma `bag-remove-repeats-filter` [\mkleeneopen{}Id\mkleeneclose{};\mkleeneopen{}reps\mkleeneclose{};\mkleeneopen{}IdDeq\mkleeneclose{}; \mkleeneopen{}\mlambda{}\msubtwo{}x.\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)\mkleeneclose{}]\mcdot{} THENA Auto ) THEN (HypSubst (-1) 0 THENA Auto) THEN (RWO "bag-remove-repeats-if-no-repeats" 0 THENA Auto) THEN (Assert \mkleeneopen{}((coeff * flrs) + 1) \mleq{} \#([x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)])\mkleeneclose{}\mcdot{} THEN 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 (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 HypSubst (-1) 0 THEN Auto) ) THEN (Assert \mkleeneopen{}\#([x\mmember{}reps|bag-deq-member(IdDeq;x;faulty)]) \mleq{} flrs\mkleeneclose{}\mcdot{} 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)\mcdot{} Home Index