Proof of Nuprl Lemma : new_23_sig

Step * 1 of Lemma new_23_sig_progress-step2

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
⊢ b_all(Id;reps;i.↓∃e':E
((e < e')
∧ (loc(e') = i ∈ Id)

                    ∧ <<<n, 0>, c>, loc(e)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(e')))

{ ((InstLemma `new_23_sig_progress-step1` [⌈Cmd⌉;⌈eq⌉;⌈reps⌉;⌈clients⌉;⌈coeff⌉;⌈flrs⌉;⌈propose⌉;⌈notify⌉;⌈slots⌉;⌈f⌉;

    ⌈es⌉;⌈e⌉;⌈n⌉;⌈c⌉;⌈faulty⌉]⋅
    THENA Auto
    )
   THEN RepeatFor 3 (ParallelLast)
   THEN UnfoldTopAb 22
   THEN RepD

   THEN (InstHyp [⌈e⌉;⌈mk-msg-interface(i;make-Msg(``new_23_sig vote``;<<<n, 0>, c>, loc(e)>))⌉] 23⋅ THENA Auto)

   THEN SquashExRepD
   THEN D 0
   THEN InstConcl [⌈e'⌉]⋅
   THEN Auto
   THEN All(RepUR ``mk-msg-interface``)
   THEN OnSomeHyp(\i.RWO "make-msg-interface-equal" i)⋅
   THEN 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. ∀i:Id. (i ↓∈ reps ⇒ make-msg-interface(0;i;make-Msg(``new_23_sig vote``;<<<n, 0>, c>, loc(e)>)) ∈ new_23_sig_main()\000C(e))
31. i : Id@i
32. i ↓∈ reps@i
33. make-msg-interface(0;i;make-Msg(``new_23_sig vote``;<<<n, 0>, c>, loc(e)>)) ∈ new_23_sig_main()(e)
34. e' : E
35. delay : ℤ
36. (e < e')
37. 0 = delay ∈ ℤ
38. i = loc(e') ∈ Id
39. make-Msg(``new_23_sig vote``;<<<n, 0>, c>, loc(e)>) = info(e') ∈ Message(f)
40. (e < e')
41. loc(e') = i ∈ Id
⊢ info(e') = mk-msg(msg-authentic(info(e'));``new_23_sig vote``;<<<n, 0>, c>, loc(e)>) ∈ 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 \mvdash{} b\_all(Id;reps;i.\mdownarrow{}\mexists{}e':E ((e < e') \mwedge{} (loc(e') = i) \mwedge{} <<<n, 0>, c>, loc(e)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(e'))) By Latex: ((InstLemma `new\_23\_sig\_progress-step1` [\mkleeneopen{}Cmd\mkleeneclose{};\mkleeneopen{}eq\mkleeneclose{};\mkleeneopen{}reps\mkleeneclose{};\mkleeneopen{}clients\mkleeneclose{};\mkleeneopen{}coeff\mkleeneclose{};\mkleeneopen{}flrs\mkleeneclose{};\mkleeneopen{}propose\mkleeneclose{}; \mkleeneopen{}notify\mkleeneclose{};\mkleeneopen{}slots\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{};\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}n\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}faulty\mkleeneclose{}]\mcdot{} THENA Auto ) THEN RepeatFor 3 (ParallelLast) THEN UnfoldTopAb 22 THEN RepD THEN (InstHyp [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}mk-msg-interface(i;make-Msg(``new\_23\_sig vote``;<<<n, 0>, c>, loc(e)>))\mkleeneclose{}] 23\mcdot{} T\000CHENA Auto) THEN SquashExRepD THEN D 0 THEN InstConcl [\mkleeneopen{}e'\mkleeneclose{}]\mcdot{} THEN Auto THEN All(RepUR ``mk-msg-interface``) THEN OnSomeHyp(\mbackslash{}i.RWO "make-msg-interface-equal" i)\mcdot{} THEN Auto THEN UnfoldAtAddr [3] 0 THEN BLemma `base-noloc-classrel-make-Msg2` THEN Auto) Home Index