Proof of Nuprl Lemma : reliable-env-property2

Step * 1 1 2 of Lemma reliable-env-property2

1. [M] : Type ─→ Type
2. Continuous+(P.M[P])
3. S : InitialSystem(P.M[P])@i
4. n2m : ℕ ─→ pMsg(P.M[P])@i
5. l2m : Id ─→ pMsg(P.M[P])@i
6. env : pEnvType(P.M[P])@i
7. pRun(S;env;n2m;l2m) ∈ pRunType(P.M[P])
8. reliable-env(env; pRun(S;env;n2m;l2m))@i
9. ∀tn:run-msg-commands(pRun(S;env;n2m;l2m))
     ∃e:runEvents(pRun(S;env;n2m;l2m))
      let t,n = tn
      in (run-info(pRun(S;env;n2m;l2m);e)

         = intransit-to-info(n2m;l2m;pRun(S;env;n2m;l2m);env;run-event-step(e);run-command(pRun(S;env;n2m;l2m);t;n))

∈ (ℤ × Id × pMsg(P.M[P])))
∧ (run-event-loc(e) = (fst(snd(run-command(pRun(S;env;n2m;l2m);t;n)))) ∈ Id)
10. t : ℕ@i
11. x : Id@i
12. \\%12 : ↑is-run-event(pRun(S;env;n2m;l2m);t;x)@i
13. P : Process(P.M[P])@i
14. (P ∈ run-event-state-when(pRun(S;env;n2m;l2m);<t, x>))@i
15. 0 < t
16. v1 : ℤ × Id@i
17. ms : pMsg(P.M[P])@i

18. let info,S = pRun(S;env;n2m;l2m) t in let ev,z,m = outl(info) in <ev, m> = <v1, ms> ∈ (ℤ × Id × pMsg(P.M[P]))@i

19. v2 : Process(P.M[P])@i
20. G : LabeledDAG(Id × pCom(P.M[P]))@i
21. Process-apply(P;ms) = <v2, G> ∈ (Process(P.M[P]) × pExt(P.M[P]))@i
22. n : ℕlg-size(G)@i
23. v3 : Id@i
24. v4 : pCom(P.M[P])@i
25. lg-label(G;n) = <v3, v4> ∈ (Id × pCom(P.M[P]))@i
26. (com-kind(v4) ∈ ``msg choose new``)@i
27. add-cause(<t, x>;G) ⊆ snd(snd((pRun(S;env;n2m;l2m) t)))
⊢ ∃e':runEvents(pRun(S;env;n2m;l2m))
((run-event-loc(e') = v3 ∈ Id)
∧ (<t, x> run-lt(pRun(S;env;n2m;l2m)) e')

   ∧ (∃n:ℕ. ∃nm:Id. ((snd(run-info(pRun(S;env;n2m;l2m);e'))) = command-to-msg(v4;n2m n;l2m nm) ∈ pMsg(P.M[P])))

   ∧ ((run-cause(pRun(S;env;n2m;l2m)) e') = (inl <t, x>) ∈ (runEvents(pRun(S;env;n2m;l2m))?)))
BY
{ ((Assert λc.fst(c) = x ∈ component(P.M[P]) ─→ 𝔹 BY
          Auto)
   THEN PromoteHyp (-1) 12
   THEN Assert ⌈∃m:ℕlg-size(snd(snd((pRun(S;env;n2m;l2m) t))))

                 (lg-label(snd(snd((pRun(S;env;n2m;l2m) t)));m) = <<t, x>, v3, v4> ∈ pInTransit(P.M[P]))⌉⋅) }

1
.....assertion.....
1. [M] : Type ─→ Type
2. Continuous+(P.M[P])
3. S : InitialSystem(P.M[P])@i
4. n2m : ℕ ─→ pMsg(P.M[P])@i
5. l2m : Id ─→ pMsg(P.M[P])@i
6. env : pEnvType(P.M[P])@i
7. pRun(S;env;n2m;l2m) ∈ pRunType(P.M[P])
8. reliable-env(env; pRun(S;env;n2m;l2m))@i
9. ∀tn:run-msg-commands(pRun(S;env;n2m;l2m))
     ∃e:runEvents(pRun(S;env;n2m;l2m))
      let t,n = tn
      in (run-info(pRun(S;env;n2m;l2m);e)

         = intransit-to-info(n2m;l2m;pRun(S;env;n2m;l2m);env;run-event-step(e);run-command(pRun(S;env;n2m;l2m);t;n))

∈ (ℤ × Id × pMsg(P.M[P])))
∧ (run-event-loc(e) = (fst(snd(run-command(pRun(S;env;n2m;l2m);t;n)))) ∈ Id)
10. t : ℕ@i
11. x : Id@i
12. λc.fst(c) = x ∈ component(P.M[P]) ─→ 𝔹
13. \\%12 : ↑is-run-event(pRun(S;env;n2m;l2m);t;x)@i
14. P : Process(P.M[P])@i
15. (P ∈ run-event-state-when(pRun(S;env;n2m;l2m);<t, x>))@i
16. 0 < t
17. v1 : ℤ × Id@i
18. ms : pMsg(P.M[P])@i

19. let info,S = pRun(S;env;n2m;l2m) t in let ev,z,m = outl(info) in <ev, m> = <v1, ms> ∈ (ℤ × Id × pMsg(P.M[P]))@i

20. v2 : Process(P.M[P])@i
21. G : LabeledDAG(Id × pCom(P.M[P]))@i
22. Process-apply(P;ms) = <v2, G> ∈ (Process(P.M[P]) × pExt(P.M[P]))@i
23. n : ℕlg-size(G)@i
24. v3 : Id@i
25. v4 : pCom(P.M[P])@i
26. lg-label(G;n) = <v3, v4> ∈ (Id × pCom(P.M[P]))@i
27. (com-kind(v4) ∈ ``msg choose new``)@i
28. add-cause(<t, x>;G) ⊆ snd(snd((pRun(S;env;n2m;l2m) t)))
⊢ ∃m:ℕlg-size(snd(snd((pRun(S;env;n2m;l2m) t))))
   (lg-label(snd(snd((pRun(S;env;n2m;l2m) t)));m) = <<t, x>, v3, v4> ∈ pInTransit(P.M[P]))

2
1. [M] : Type ─→ Type
2. Continuous+(P.M[P])
3. S : InitialSystem(P.M[P])@i
4. n2m : ℕ ─→ pMsg(P.M[P])@i
5. l2m : Id ─→ pMsg(P.M[P])@i
6. env : pEnvType(P.M[P])@i
7. pRun(S;env;n2m;l2m) ∈ pRunType(P.M[P])
8. reliable-env(env; pRun(S;env;n2m;l2m))@i
9. ∀tn:run-msg-commands(pRun(S;env;n2m;l2m))
     ∃e:runEvents(pRun(S;env;n2m;l2m))
      let t,n = tn
      in (run-info(pRun(S;env;n2m;l2m);e)

         = intransit-to-info(n2m;l2m;pRun(S;env;n2m;l2m);env;run-event-step(e);run-command(pRun(S;env;n2m;l2m);t;n))

19. let info,S = pRun(S;env;n2m;l2m) t in let ev,z,m = outl(info) in <ev, m> = <v1, ms> ∈ (ℤ × Id × pMsg(P.M[P]))@i

   ∧ (∃n:ℕ. ∃nm:Id. ((snd(run-info(pRun(S;env;n2m;l2m);e'))) = command-to-msg(v4;n2m n;l2m nm) ∈ pMsg(P.M[P])))

∧ ((run-cause(pRun(S;env;n2m;l2m)) e') = (inl <t, x>) ∈ (runEvents(pRun(S;env;n2m;l2m))?)))

Latex:

Latex:
1. [M] : Type {}\mrightarrow{} Type 2. Continuous+(P.M[P]) 3. S : InitialSystem(P.M[P])@i 4. n2m : \mBbbN{} {}\mrightarrow{} pMsg(P.M[P])@i 5. l2m : Id {}\mrightarrow{} pMsg(P.M[P])@i 6. env : pEnvType(P.M[P])@i 7. pRun(S;env;n2m;l2m) \mmember{} pRunType(P.M[P]) 8. reliable-env(env; pRun(S;env;n2m;l2m))@i 9. \mforall{}tn:run-msg-commands(pRun(S;env;n2m;l2m)) \mexists{}e:runEvents(pRun(S;env;n2m;l2m)) let t,n = tn in ... \mwedge{} (run-event-loc(e) = (fst(snd(run-command(pRun(S;env;n2m;l2m);t;n))))) 10. t : \mBbbN{}@i 11. x : Id@i 12. \mbackslash{}\mbackslash{}\%12 : \muparrow{}is-run-event(pRun(S;env;n2m;l2m);t;x)@i 13. P : Process(P.M[P])@i 14. (P \mmember{} run-event-state-when(pRun(S;env;n2m;l2m);<t, x>))@i 15. 0 < t 16. v1 : \mBbbZ{} \mtimes{} Id@i 17. ms : pMsg(P.M[P])@i 18. let info,S = pRun(S;env;n2m;l2m) t in let ev,z,m = outl(info) in <ev, m> = <v1, ms>@i 19. v2 : Process(P.M[P])@i 20. G : LabeledDAG(Id \mtimes{} pCom(P.M[P]))@i 21. Process-apply(P;ms) = <v2, G>@i 22. n : \mBbbN{}lg-size(G)@i 23. v3 : Id@i 24. v4 : pCom(P.M[P])@i 25. lg-label(G;n) = <v3, v4>@i 26. (com-kind(v4) \mmember{} ``msg choose new``)@i 27. add-cause(<t, x>G) \msubseteq{} snd(snd((pRun(S;env;n2m;l2m) t))) \mvdash{} \mexists{}e':runEvents(pRun(S;env;n2m;l2m)) ((run-event-loc(e') = v3) \mwedge{} (<t, x> run-lt(pRun(S;env;n2m;l2m)) e') \mwedge{} (\mexists{}n:\mBbbN{}. \mexists{}nm:Id. ((snd(run-info(pRun(S;env;n2m;l2m);e'))) = command-to-msg(v4;n2m n;l2m nm))) \mwedge{} ((run-cause(pRun(S;env;n2m;l2m)) e') = (inl <t, x>))) By Latex: ((Assert \mlambda{}c.fst(c) = x \mmember{} component(P.M[P]) {}\mrightarrow{} \mBbbB{} BY Auto) THEN PromoteHyp (-1) 12 THEN Assert \mkleeneopen{}\mexists{}m:\mBbbN{}lg-size(snd(snd((pRun(S;env;n2m;l2m) t)))) (lg-label(snd(snd((pRun(S;env;n2m;l2m) t)));m) = <<t, x>, v3, v4>)\mkleeneclose{}\mcdot{}) Home Index