Step
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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. λc.fst(c) = x ∈ component(P.M[P]) ─→ 𝔹
13. ↑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)))
29. m : ℕlg-size(snd(snd((pRun(S;env;n2m;l2m) t))))
30. lg-label(snd(snd((pRun(S;env;n2m;l2m) t)));m) = <<t, x>, v3, v4> ∈ pInTransit(P.M[P])
31. t1 : ℕ
32. v5 : ℤ × Id × Id × pMsg(P.M[P])?@i
33. v7 : Top@i
34. v8 : LabeledDAG(pInTransit(P.M[P]))@i
35. (pRun(S;env;n2m;l2m) t1) = <v5, v7, v8> ∈ (ℤ × Id × Id × pMsg(P.M[P])? × Top × LabeledDAG(pInTransit(P.M[P])))@i
36. n1 : ℕ
37. n1 < lg-size(snd(snd(<v5, v7, v8>)))
⊢ com-kind(snd(snd(lg-label(v8;n1)))) ∈ Atom
BY
{ (GenConclTerm ⌈lg-label(v8;n1)⌉⋅ THEN Auto) }
1
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. ↑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)))
29. m : ℕlg-size(snd(snd((pRun(S;env;n2m;l2m) t))))
30. lg-label(snd(snd((pRun(S;env;n2m;l2m) t)));m) = <<t, x>, v3, v4> ∈ pInTransit(P.M[P])
31. t1 : ℕ
32. v5 : ℤ × Id × Id × pMsg(P.M[P])?@i
33. v7 : Top@i
34. v8 : LabeledDAG(pInTransit(P.M[P]))@i
35. (pRun(S;env;n2m;l2m) t1) = <v5, v7, v8> ∈ (ℤ × Id × Id × pMsg(P.M[P])? × Top × LabeledDAG(pInTransit(P.M[P])))@i
36. n1 : ℕ
37. n1 < lg-size(snd(snd(<v5, v7, v8>)))
38. v : pInTransit(P.M[P])@i
39. lg-label(v8;n1) = v ∈ pInTransit(P.M[P])@i
⊢ com-kind(snd(snd(v))) ∈ Atom
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. \mlambda{}c.fst(c) = x \mmember{} component(P.M[P]) {}\mrightarrow{} \mBbbB{}
13. \muparrow{}is-run-event(pRun(S;env;n2m;l2m);t;x)@i
14. P : Process(P.M[P])@i
15. (P \mmember{} run-event-state-when(pRun(S;env;n2m;l2m);<t, x>))@i
16. 0 < t
17. v1 : \mBbbZ{} \mtimes{} 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>@i
20. v2 : Process(P.M[P])@i
21. G : LabeledDAG(Id \mtimes{} pCom(P.M[P]))@i
22. Process-apply(P;ms) = <v2, G>@i
23. n : \mBbbN{}lg-size(G)@i
24. v3 : Id@i
25. v4 : pCom(P.M[P])@i
26. lg-label(G;n) = <v3, v4>@i
27. (com-kind(v4) \mmember{} ``msg choose new``)@i
28. add-cause(<t, x>G) \msubseteq{} snd(snd((pRun(S;env;n2m;l2m) t)))
29. m : \mBbbN{}lg-size(snd(snd((pRun(S;env;n2m;l2m) t))))
30. lg-label(snd(snd((pRun(S;env;n2m;l2m) t)));m) = <<t, x>, v3, v4>
31. t1 : \mBbbN{}
32. v5 : \mBbbZ{} \mtimes{} Id \mtimes{} Id \mtimes{} pMsg(P.M[P])?@i
33. v7 : Top@i
34. v8 : LabeledDAG(pInTransit(P.M[P]))@i
35. (pRun(S;env;n2m;l2m) t1) = <v5, v7, v8>@i
36. n1 : \mBbbN{}
37. n1 < lg-size(snd(snd(<v5, v7, v8>)))
\mvdash{} com-kind(snd(snd(lg-label(v8;n1)))) \mmember{} Atom
By
Latex:
(GenConclTerm \mkleeneopen{}lg-label(v8;n1)\mkleeneclose{}\mcdot{} THEN Auto)
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