Proof of Nuprl Lemma : reliable-env-property

Step * 1 1 of Lemma reliable-env-property

1. [M] : Type ─→ Type
2. Continuous+(P.M[P])
3. S0 : 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(S0;env;n2m;l2m) ∈ pRunType(P.M[P])
8. reliable-env(env; pRun(S0;env;n2m;l2m))@i
9. ∀tn:run-msg-commands(pRun(S0;env;n2m;l2m))
     let t,n = tn
     in ∃t':ℕ
         ((t < t'
         c∧ (((fst((env t' pRun(S0;env;n2m;l2m)))) ≤ n)

            ∧ ↑lg-is-source(run-intransit(pRun(S0;env;n2m;l2m);t');fst((env t' pRun(S0;env;n2m;l2m))))

              supposing 0 < lg-size(run-intransit(pRun(S0;env;n2m;l2m);t'))))
         ∧ (∀i:ℕ
              ¬(t < i
              c∧ (((fst((env i pRun(S0;env;n2m;l2m)))) ≤ n)

                 ∧ ↑lg-is-source(run-intransit(pRun(S0;env;n2m;l2m);i);fst((env i pRun(S0;env;n2m;l2m))))

                   supposing 0 < lg-size(run-intransit(pRun(S0;env;n2m;l2m);i))))
              supposing i < t'))
10. n : ℕ@i
11. ∀n1:ℕn. ∀t:ℕ.
      ((run-command-node(pRun(S0;env;n2m;l2m);t;n1)
      ∧ (com-kind(snd(snd(run-command(pRun(S0;env;n2m;l2m);t;n1)))) ∈ ``msg choose new``))
      ⇒ (∃e:runEvents(pRun(S0;env;n2m;l2m))
           let t,n = <t, n1>
           in (run-info(pRun(S0;env;n2m;l2m);e)

              = intransit-to-info(n2m;l2m;pRun(S0;env;n2m;l2m);env;run-event-step(e);run-command(...;t;n))

              ∈ (ℤ × Id × pMsg(P.M[P])))
              ∧ (run-event-loc(e) = (fst(snd(run-command(pRun(S0;env;n2m;l2m);t;n)))) ∈ Id)))@i
12. t : ℕ@i
13. run-command-node(pRun(S0;env;n2m;l2m);t;n)@i
14. (com-kind(snd(snd(run-command(pRun(S0;env;n2m;l2m);t;n)))) ∈ ``msg choose new``)@i
15. let t,n = <t, n>
    in ∃t':ℕ
        ((t < t'
        c∧ (((fst((env t' pRun(S0;env;n2m;l2m)))) ≤ n)

           ∧ ↑lg-is-source(run-intransit(pRun(S0;env;n2m;l2m);t');fst((env t' pRun(S0;env;n2m;l2m))))

             supposing 0 < lg-size(run-intransit(pRun(S0;env;n2m;l2m);t'))))
        ∧ (∀i:ℕ
             ¬(t < i
             c∧ (((fst((env i pRun(S0;env;n2m;l2m)))) ≤ n)

                ∧ ↑lg-is-source(run-intransit(pRun(S0;env;n2m;l2m);i);fst((env i pRun(S0;env;n2m;l2m))))

                  supposing 0 < lg-size(run-intransit(pRun(S0;env;n2m;l2m);i))))
             supposing i < t'))
⊢ ∃e:runEvents(pRun(S0;env;n2m;l2m))
   let t,n = <t, n>
   in (run-info(pRun(S0;env;n2m;l2m);e)

      = intransit-to-info(n2m;l2m;pRun(S0;env;n2m;l2m);env;run-event-step(e);run-command(pRun(S0;env;n2m;l2m);t;n))

      ∈ (ℤ × Id × pMsg(P.M[P])))
      ∧ (run-event-loc(e) = (fst(snd(run-command(pRun(S0;env;n2m;l2m);t;n)))) ∈ Id)
BY
{ (Thin 9 THEN Reduce (-1) THEN ExRepD)⋅ }

1
1. [M] : Type ─→ Type
2. Continuous+(P.M[P])
3. S0 : 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(S0;env;n2m;l2m) ∈ pRunType(P.M[P])
8. reliable-env(env; pRun(S0;env;n2m;l2m))@i
9. n : ℕ@i
10. ∀n1:ℕn. ∀t:ℕ.
      ((run-command-node(pRun(S0;env;n2m;l2m);t;n1)
      ∧ (com-kind(snd(snd(run-command(pRun(S0;env;n2m;l2m);t;n1)))) ∈ ``msg choose new``))
      ⇒ (∃e:runEvents(pRun(S0;env;n2m;l2m))
           let t,n = <t, n1>
           in (run-info(pRun(S0;env;n2m;l2m);e)

              = intransit-to-info(n2m;l2m;pRun(S0;env;n2m;l2m);env;run-event-step(e);run-command(...;t;n))

              ∈ (ℤ × Id × pMsg(P.M[P])))
              ∧ (run-event-loc(e) = (fst(snd(run-command(pRun(S0;env;n2m;l2m);t;n)))) ∈ Id)))@i
11. t : ℕ@i
12. run-command-node(pRun(S0;env;n2m;l2m);t;n)@i
13. (com-kind(snd(snd(run-command(pRun(S0;env;n2m;l2m);t;n)))) ∈ ``msg choose new``)@i
14. t' : ℕ
15. t < t'
16. (fst((env t' pRun(S0;env;n2m;l2m)))) ≤ n
17. ↑lg-is-source(run-intransit(pRun(S0;env;n2m;l2m);t');fst((env t' pRun(S0;env;n2m;l2m))))
    supposing 0 < lg-size(run-intransit(pRun(S0;env;n2m;l2m);t'))
18. ∀i:ℕ
      ¬(t < i
      c∧ (((fst((env i pRun(S0;env;n2m;l2m)))) ≤ n)
         ∧ ↑lg-is-source(run-intransit(pRun(S0;env;n2m;l2m);i);fst((env i pRun(S0;env;n2m;l2m))))
           supposing 0 < lg-size(run-intransit(pRun(S0;env;n2m;l2m);i))))
      supposing i < t'
⊢ ∃e:runEvents(pRun(S0;env;n2m;l2m))
   let t,n = <t, n>
   in (run-info(pRun(S0;env;n2m;l2m);e)

      = intransit-to-info(n2m;l2m;pRun(S0;env;n2m;l2m);env;run-event-step(e);run-command(pRun(S0;env;n2m;l2m);t;n))

∈ (ℤ × Id × pMsg(P.M[P])))
∧ (run-event-loc(e) = (fst(snd(run-command(pRun(S0;env;n2m;l2m);t;n)))) ∈ Id)

Latex:

Latex:
1. [M] : Type {}\mrightarrow{} Type 2. Continuous+(P.M[P]) 3. S0 : 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(S0;env;n2m;l2m) \mmember{} pRunType(P.M[P]) 8. reliable-env(env; pRun(S0;env;n2m;l2m))@i 9. \mforall{}tn:run-msg-commands(pRun(S0;env;n2m;l2m)) let t,n = tn in \mexists{}t':\mBbbN{} ((t < t' c\mwedge{} (((fst((env t' pRun(S0;env;n2m;l2m)))) \mleq{} n) \mwedge{} \muparrow{}lg-is-source(run-intransit(pRun(S0;env;n2m;l2m);t');fst((env t' pRun(S0;env;n2m;l2m)))) supposing 0 < lg-size(run-intransit(pRun(S0;env;n2m;l2m);t')))) \mwedge{} (\mforall{}i:\mBbbN{} \mneg{}(t < i c\mwedge{} (((fst((env i pRun(S0;env;n2m;l2m)))) \mleq{} n) \mwedge{} \muparrow{}lg-is-source(run-intransit(pRun(S0;env;n2m;l2m);i);fst((env i pRun(S0;env;n2m;l2m)))) supposing 0 < lg-size(run-intransit(pRun(S0;env;n2m;l2m);i)))) supposing i < t')) 10. n : \mBbbN{}@i 11. \mforall{}n1:\mBbbN{}n. \mforall{}t:\mBbbN{}. ((run-command-node(pRun(S0;env;n2m;l2m);t;n1) \mwedge{} (com-kind(snd(snd(run-command(pRun(S0;env;n2m;l2m);t;n1)))) \mmember{} ``msg choose new``)) {}\mRightarrow{} (\mexists{}e:runEvents(pRun(S0;env;n2m;l2m)) let t,n = <t, n1> in (run-info(pRun(S0;env;n2m;l2m);e) = intransit-to-info(n2m;l2m;pRun(S0;env;n2m;l2m);env;run-event-step(e);...)) \mwedge{} (run-event-loc(e) = (fst(snd(run-command(pRun(S0;env;n2m;l2m);t;n)))))))@i 12. t : \mBbbN{}@i 13. run-command-node(pRun(S0;env;n2m;l2m);t;n)@i 14. (com-kind(snd(snd(run-command(pRun(S0;env;n2m;l2m);t;n)))) \mmember{} ``msg choose new``)@i 15. let t,n = <t, n> in \mexists{}t':\mBbbN{} ((t < t' c\mwedge{} (((fst((env t' pRun(S0;env;n2m;l2m)))) \mleq{} n) \mwedge{} \muparrow{}lg-is-source(run-intransit(pRun(S0;env;n2m;l2m);t');fst((env t' pRun(S0;env;n2m;l2m)))) supposing 0 < lg-size(run-intransit(pRun(S0;env;n2m;l2m);t')))) \mwedge{} (\mforall{}i:\mBbbN{} \mneg{}(t < i c\mwedge{} (((fst((env i pRun(S0;env;n2m;l2m)))) \mleq{} n) \mwedge{} \muparrow{}lg-is-source(run-intransit(pRun(S0;env;n2m;l2m);i);fst((env i pRun(S0;env;n2m;l2m)))) supposing 0 < lg-size(run-intransit(pRun(S0;env;n2m;l2m);i)))) supposing i < t')) \mvdash{} \mexists{}e:runEvents(pRun(S0;env;n2m;l2m)) let t,n = <t, n> in (run-info(pRun(S0;env;n2m;l2m);e) = intransit-to-info(n2m;l2m;pRun(S0;env;n2m;l2m);env;run-event-step(e);run-command(...;t;n))) \mwedge{} (run-event-loc(e) = (fst(snd(run-command(pRun(S0;env;n2m;l2m);t;n))))) By Latex: (Thin 9 THEN Reduce (-1) THEN ExRepD)\mcdot{} Home Index