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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. ∀i:ℕ
¬(t < i
c∧ (((fst((env i pRun(S0;env;n2m;l2m)))) ≤ n)
∧ ↑lg-is-source(snd(snd((pRun(S0;env;n2m;l2m) (i - 1))));fst((env i pRun(S0;env;n2m;l2m))))
supposing 0 < lg-size(snd(snd((pRun(S0;env;n2m;l2m) (i - 1)))))))
supposing i < t'
18. ∀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]))))))
19. n < lg-size(snd(snd((pRun(S0;env;n2m;l2m) (t' - 1)))))
20. ∀x:ℕ
((x ≤ n)
⇒ (run-command(pRun(S0;env;n2m;l2m);t' - 1;x) = run-command(pRun(S0;env;n2m;l2m);t;x) ∈ pInTransit(P.M[P])))
21. run-command(pRun(S0;env;n2m;l2m);t' - 1;n) = run-command(pRun(S0;env;n2m;l2m);t;n) ∈ pInTransit(P.M[P])
22. fst((env t' pRun(S0;env;n2m;l2m))) < n
23. ↑lg-is-source(snd(snd((pRun(S0;env;n2m;l2m) (t' - 1))));fst((env t' pRun(S0;env;n2m;l2m))))
24. (com-kind(snd(snd(run-command(pRun(S0;env;n2m;l2m);t' - 1;n)))) ∈ ``msg choose new``)
⊢ run-command-node(pRun(S0;env;n2m;l2m);t';n - 1)
∧ (run-command(pRun(S0;env;n2m;l2m);t' - 1;n) = run-command(pRun(S0;env;n2m;l2m);t';n - 1) ∈ pInTransit(P.M[P]))
BY
{ (RepeatFor 3 (MoveToConcl (-1))
THEN MoveToConcl (-3)
THEN RepUR ``run-command-node run-command`` 0
THEN (Subst' pRun(S0;env;n2m;l2m) t' ~ let n,m,nm = env t' pRun(S0;env;n2m;l2m) in
do-chosen-command(n2m;l2m;t';snd((pRun(S0;env;n2m;l2m) (t' - 1)));n;m;nm) 0
THENA (RW (AddrC [1] (RecUnfoldC `pRun`)) 0 THEN Reduce 0 THEN AutoSplit)⋅
))⋅ }
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. ∀i:ℕ
¬(t < i
c∧ (((fst((env i pRun(S0;env;n2m;l2m)))) ≤ n)
∧ ↑lg-is-source(snd(snd((pRun(S0;env;n2m;l2m) (i - 1))));fst((env i pRun(S0;env;n2m;l2m))))
supposing 0 < lg-size(snd(snd((pRun(S0;env;n2m;l2m) (i - 1)))))))
supposing i < t'
18. ∀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]))))))
19. ∀x:ℕ
((x ≤ n)
⇒ (run-command(pRun(S0;env;n2m;l2m);t' - 1;x) = run-command(pRun(S0;env;n2m;l2m);t;x) ∈ pInTransit(P.M[P])))
20. run-command(pRun(S0;env;n2m;l2m);t' - 1;n) = run-command(pRun(S0;env;n2m;l2m);t;n) ∈ pInTransit(P.M[P])
⊢ n < lg-size(snd(snd((pRun(S0;env;n2m;l2m) (t' - 1)))))
⇒ fst((env t' pRun(S0;env;n2m;l2m))) < n
⇒ (↑lg-is-source(snd(snd((pRun(S0;env;n2m;l2m) (t' - 1))));fst((env t' pRun(S0;env;n2m;l2m)))))
⇒ (com-kind(snd(snd(lg-label(snd(snd((pRun(S0;env;n2m;l2m) (t' - 1))));n)))) ∈ ``msg choose new``)
⇒ (n - 1 < lg-size(snd(snd(let n,m,nm = env t' pRun(S0;env;n2m;l2m) in
do-chosen-command(n2m;l2m;t';snd((pRun(S0;env;n2m;l2m) (t' - 1)));n;m;nm))))
∧ (lg-label(snd(snd((pRun(S0;env;n2m;l2m) (t' - 1))));n)
= lg-label(snd(snd(let n,m,nm = env t' pRun(S0;env;n2m;l2m) in
do-chosen-command(n2m;l2m;t';snd((pRun(S0;env;n2m;l2m) (t' - 1)));n;m;nm)));n - 1)
∈ pInTransit(P.M[P])))
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. \mforall{}i:\mBbbN{}
\mneg{}(t < i
c\mwedge{} (((fst((env i pRun(S0;env;n2m;l2m)))) \mleq{} n)
\mwedge{} \muparrow{}lg-is-source(snd(snd((pRun(S0;env;n2m;l2m) (i - 1))));fst((env i
pRun(S0;env;n2m;l2m))))
supposing 0 < lg-size(snd(snd((pRun(S0;env;n2m;l2m) (i - 1)))))))
supposing i < t'
18. \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))))))
19. n < lg-size(snd(snd((pRun(S0;env;n2m;l2m) (t' - 1)))))
20. \mforall{}x:\mBbbN{}
((x \mleq{} n)
{}\mRightarrow{} (run-command(pRun(S0;env;n2m;l2m);t' - 1;x) = run-command(pRun(S0;env;n2m;l2m);t;x)))
21. run-command(pRun(S0;env;n2m;l2m);t' - 1;n) = run-command(pRun(S0;env;n2m;l2m);t;n)
22. fst((env t' pRun(S0;env;n2m;l2m))) < n
23. \muparrow{}lg-is-source(snd(snd((pRun(S0;env;n2m;l2m) (t' - 1))));fst((env t' pRun(S0;env;n2m;l2m))))
24. (com-kind(snd(snd(run-command(pRun(S0;env;n2m;l2m);t' - 1;n)))) \mmember{} ``msg choose new``)
\mvdash{} run-command-node(pRun(S0;env;n2m;l2m);t';n - 1)
\mwedge{} (run-command(pRun(S0;env;n2m;l2m);t' - 1;n) = run-command(pRun(S0;env;n2m;l2m);t';n - 1))
By
Latex:
(RepeatFor 3 (MoveToConcl (-1))
THEN MoveToConcl (-3)
THEN RepUR ``run-command-node run-command`` 0
THEN (Subst' pRun(S0;env;n2m;l2m) t' \msim{} let n,m,nm = env t' pRun(S0;env;n2m;l2m) in
do-chosen-command(n2m;l2m;t';snd((pRun(S0;env;n2m;l2m) (t' - 1)));n;m;nm) 0
THENA (RW (AddrC [1] (RecUnfoldC `pRun`)) 0 THEN Reduce 0 THEN AutoSplit)\mcdot{}
))\mcdot{}
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