Proof of Nuprl Lemma : new_23_sig

Step * 1 2 2 1 of Lemma new_23_sig_progress-step10

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. e' : E
30. ((coeff * flrs) + 1)
= ||mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
              λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
              [e, e'])||
∈ ℤ
31. e ≤loc e'
32. ↑new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0)
33. ¬↑(∃x∈[e, e'].new_23_sig_decided_with_id(Cmd;notify;propose;f;es;x;n))_b
34. e1 : E
35. c' : Cmd
36. e ≤loc e1
37. mk-msg-interface(loc(e);make-Msg(``new_23_sig decided``;<n, c'>)) ∈ new_23_sig_main()(e1)
⊢ ↓∃e':E
    ∃c':Cmd
     ((e <loc e')
     ∧ (<n, c'> ∈ new_23_sig_decided'base(Cmd;notify;propose;f)(e')
       ∨ (∃r:ℕ⁺. <<n, r>, c'> ∈ new_23_sig_retry'base(Cmd;notify;propose;f)(e'))))
BY
{ (UnfoldTopAb 22
   THEN RepD

   THEN (InstHyp [⌜e1⌝;⌜mk-msg-interface(loc(e);make-Msg(``new_23_sig decided``;<n, c'>))⌝] 23⋅ THENA Auto)

   THEN SquashExRepD
   THEN All(RepUR ``mk-msg-interface``)
   THEN OnSomeHyp(\i.RWO "make-msg-interface-equal" i)⋅
   THEN Auto
   THEN D 0
   THEN InstConcl [⌜e2⌝;⌜c'⌝]⋅
   THEN Auto
   THEN (OrLeft THENA Auto)
   THEN UnfoldAtAddr [3] 0
   THEN BLemma `base-noloc-classrel-make-Msg2`
   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. #(faulty) ≤ flrs@i
23. ∀e:E. ∀dmsg:Interface.
      (dmsg ∈ new_23_sig_main()(e)
      ⇒ (¬loc(e) ↓∈ faulty)
      ⇒ loc(e) ↓∈ reps
      ⇒

 (↓∃e':E. ∃delay:ℤ. ((e < e') ∧ (dmsg = make-msg-interface(delay;loc(e');info(e')) ∈ Interface))))@i

24. bag-no-repeats(Id;reps)@i
25. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
26. loc(e) ↓∈ reps@i
27. ¬loc(e) ↓∈ faulty@i
28. <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)@i
29. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
30. e' : E
31. ((coeff * flrs) + 1)
= ||mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
[e, e'])||
∈ ℤ
32. e ≤loc e'
33. ↑new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0)
34. ¬↑(∃x∈[e, e'].new_23_sig_decided_with_id(Cmd;notify;propose;f;es;x;n))_b
35. e1 : E
36. c' : Cmd
37. e ≤loc e1
38. make-msg-interface(0;loc(e);make-Msg(``new_23_sig decided``;<n, c'>)) ∈ new_23_sig_main()(e1)
39. e2 : E
40. delay : ℤ
41. (e1 < e2)
42. 0 = delay ∈ ℤ
43. loc(e) = loc(e2) ∈ Id
44. make-Msg(``new_23_sig decided``;<n, c'>) = info(e2) ∈ Message(f)
45. (e <loc e2)
⊢ info(e2) = mk-msg(msg-authentic(info(e2));``new_23_sig decided``;<n, c'>) ∈ Message(f)

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. e' : E 30. ((coeff * flrs) + 1) = ||mapfilter(\mlambda{}e.<snd(msgval(e)), e, snd(fst(msgval(e)))> \mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0); [e, e'])|| 31. e \mleq{}loc e' 32. \muparrow{}new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0) 33. \mneg{}\muparrow{}(\mexists{}x\mmember{}[e, e'].new\_23\_sig\_decided\_with\_id(Cmd;notify;propose;f;es;x;n))\_b 34. e1 : E 35. c' : Cmd 36. e \mleq{}loc e1 37. mk-msg-interface(loc(e);make-Msg(``new\_23\_sig decided``;<n, c'>)) \mmember{} new\_23\_sig\_main()(e1) \mvdash{} \mdownarrow{}\mexists{}e':E \mexists{}c':Cmd ((e <loc e') \mwedge{} (<n, c'> \mmember{} new\_23\_sig\_decided'base(Cmd;notify;propose;f)(e') \mvee{} (\mexists{}r:\mBbbN{}\msupplus{}. <<n, r>, c'> \mmember{} new\_23\_sig\_retry'base(Cmd;notify;propose;f)(e')))) By Latex: (UnfoldTopAb 22 THEN RepD THEN (InstHyp [\mkleeneopen{}e1\mkleeneclose{};\mkleeneopen{}mk-msg-interface(loc(e);make-Msg(``new\_23\_sig decided``;<n, c'>))\mkleeneclose{}] 23\mcdot{} THENA \000CAuto) THEN SquashExRepD THEN All(RepUR ``mk-msg-interface``) THEN OnSomeHyp(\mbackslash{}i.RWO "make-msg-interface-equal" i)\mcdot{} THEN Auto THEN D 0 THEN InstConcl [\mkleeneopen{}e2\mkleeneclose{};\mkleeneopen{}c'\mkleeneclose{}]\mcdot{} THEN Auto THEN (OrLeft THENA Auto) THEN UnfoldAtAddr [3] 0 THEN BLemma `base-noloc-classrel-make-Msg2` THEN Auto) Home Index