---

Step * 1 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. 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)
BY
{ Assert ⌈∀t2:ℕ
            ((t ≤ t2)
            ⇒ t2 < t'
            ⇒ (n < lg-size(snd(snd((pRun(S0;env;n2m;l2m) t2))))
               ∧ (∀x:ℕ
                    ((x ≤ n)
                    ⇒ (run-command(pRun(S0;env;n2m;l2m);t2;x)
                       = run-command(pRun(S0;env;n2m;l2m);t;x)
                       ∈ pInTransit(P.M[P]))))))⌉⋅ }

1
.....assertion..... 
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'
⊢ ∀t2:ℕ
    ((t ≤ t2)
    ⇒ t2 < t'
    ⇒ (n < lg-size(snd(snd((pRun(S0;env;n2m;l2m) t2))))
       ∧ (∀x:ℕ
            ((x ≤ n)
            ⇒ (run-command(pRun(S0;env;n2m;l2m);t2;x) = run-command(pRun(S0;env;n2m;l2m);t;x) ∈ pInTransit(P.M[P]))))))

2
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'
19. ∀t2:ℕ
      ((t ≤ t2)
      ⇒ t2 < t'
      ⇒ (n < lg-size(snd(snd((pRun(S0;env;n2m;l2m) t2))))
         ∧ (∀x:ℕ
              ((x ≤ n)
              ⇒ (run-command(pRun(S0;env;n2m;l2m);t2;x)
                 = run-command(pRun(S0;env;n2m;l2m);t;x)
                 ∈ pInTransit(P.M[P]))))))
⊢ ∃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.  n  :  \mBbbN{}@i
10.  \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
11.  t  :  \mBbbN{}@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))))  \mmember{}  ``msg  choose  new``)@i
14.  t'  :  \mBbbN{}
15.  t  <  t'
16.  (fst((env  t'  pRun(S0;env;n2m;l2m))))  \mleq{}  n
17.  \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'))
18.  \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:
Assert  \mkleeneopen{}\mforall{}t2:\mBbbN{}
                    ((t  \mleq{}  t2)
                    {}\mRightarrow{}  t2  <  t'
                    {}\mRightarrow{}  (n  <  lg-size(snd(snd((pRun(S0;env;n2m;l2m)  t2))))
                          \mwedge{}  (\mforall{}x:\mBbbN{}
                                    ((x  \mleq{}  n)
                                    {}\mRightarrow{}  (run-command(pRun(S0;env;n2m;l2m);t2;x)
                                          =  run-command(pRun(S0;env;n2m;l2m);t;x))))))\mkleeneclose{}\mcdot{}

---

Home Index