Step
*
1
1
1
2
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'
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)
BY
{ ((InstHyp [⌜t' - 1⌝] (-1)⋅ THENA Auto)
THEN D -1
THEN (InstHyp [⌜n⌝] (-1)⋅ THENA Auto)
THEN (RevHypSubst' (-1) 0 THENA (Auto THEN (MemTypeCD THEN Auto) THEN BLemma `run-event-step-positive` THEN Auto))
THEN (Decide ⌜fst((env t' pRun(S0;env;n2m;l2m))) < n⌝⋅ THENA (Auto THEN DoSubsume⋅ THEN Auto))) }
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'
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]))))))
20. n < lg-size(snd(snd((pRun(S0;env;n2m;l2m) (t' - 1)))))
21. ∀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])))
22. run-command(pRun(S0;env;n2m;l2m);t' - 1;n) = run-command(pRun(S0;env;n2m;l2m);t;n) ∈ pInTransit(P.M[P])
23. fst((env t' pRun(S0;env;n2m;l2m))) < n
⊢ ∃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)
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]))))))
20. n < lg-size(snd(snd((pRun(S0;env;n2m;l2m) (t' - 1)))))
21. ∀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])))
22. run-command(pRun(S0;env;n2m;l2m);t' - 1;n) = run-command(pRun(S0;env;n2m;l2m);t;n) ∈ pInTransit(P.M[P])
23. ¬fst((env t' pRun(S0;env;n2m;l2m))) < n
⊢ ∃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'
19. \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))))))
\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:
((InstHyp [\mkleeneopen{}t' - 1\mkleeneclose{}] (-1)\mcdot{} THENA Auto)
THEN D -1
THEN (InstHyp [\mkleeneopen{}n\mkleeneclose{}] (-1)\mcdot{} THENA Auto)
THEN (RevHypSubst' (-1) 0
THENA (Auto THEN (MemTypeCD THEN Auto) THEN BLemma `run-event-step-positive` THEN Auto)
)
THEN (Decide \mkleeneopen{}fst((env t' pRun(S0;env;n2m;l2m))) < n\mkleeneclose{}\mcdot{} THENA (Auto THEN DoSubsume\mcdot{} THEN Auto)))
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