Step
*
1
1
1
1
1
1
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. reliable-env(env; pRun(S;env;n2m;l2m))@i
8. ∀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)
9. t : {1...}
10. x : Id@i
11. [%12] : ↑is-run-event(pRun(S;env;n2m;l2m);t;x)@i
12. P : Process(P.M[P])@i
13. (P ∈ run-event-state-when(pRun(S;env;n2m;l2m);<t, x>))@i
14. 0 < t
15. v1 : ℤ × Id@i
16. ms : pMsg(P.M[P])@i
17. v2 : Process(P.M[P])@i
18. G : LabeledDAG(Id × pCom(P.M[P]))@i
19. Process-apply(P;ms) = <v2, G> ∈ (Process(P.M[P]) × pExt(P.M[P]))@i
20. n : ℕlg-size(G)@i
21. v3 : Id@i
22. v4 : pCom(P.M[P])@i
23. lg-label(G;n) = <v3, v4> ∈ (Id × pCom(P.M[P]))@i
24. (com-kind(v4) ∈ ``msg choose new``)@i
25. v5 : ℤ × Id × Id × pMsg(P.M[P])?@i
26. v7 : component(P.M[P]) List@i
27. v8 : LabeledDAG(pInTransit(P.M[P]))@i
28. (pRun(S;env;n2m;l2m) (t - 1)) = <v5, v7, v8> ∈ (ℤ × Id × Id × pMsg(P.M[P])? × System(P.M[P]))@i
29. v9 : ℕ@i
30. v11 : ℕ@i
31. v12 : Id@i
32. (env t pRun(S;env;n2m;l2m)) = <v9, v11, v12> ∈ (ℕ × ℕ × Id)@i
33. ↑lg-is-source(v8;v9)
⊢ (P ∈ mapfilter(λc.(snd(c));λc.fst(c) = x;v7))
⇒ (↑let info,S = let ev,x,c = lg-label(v8;v9) in
if com-kind(c) =a "msg" then <inl <ev, x, comm-msg(c)>, deliver-msg(t;comm-msg(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "create" then <inr ⋅ , create-component(t;comm-create(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "choose" then <inl <ev, x, n2m v11>, deliver-msg(t;n2m v11;x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "new" then <inl <ev, x, l2m v12>, deliver-msg(t;l2m v12;x;v7;lg-remove(v8;v9))>
else <inr ⋅ , v7, lg-remove(v8;v9)>
fi
in isl(info) ∧b let ev,z,m = outl(info) in x = z)
⇒ (let info,S = let ev,x,c = lg-label(v8;v9) in
if com-kind(c) =a "msg" then <inl <ev, x, comm-msg(c)>, deliver-msg(t;comm-msg(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "create" then <inr ⋅ , create-component(t;comm-create(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "choose" then <inl <ev, x, n2m v11>, deliver-msg(t;n2m v11;x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "new" then <inl <ev, x, l2m v12>, deliver-msg(t;l2m v12;x;v7;lg-remove(v8;v9))>
else <inr ⋅ , v7, lg-remove(v8;v9)>
fi
in let ev,z,m = outl(info) in
<ev, m>
= <v1, ms>
∈ (ℤ × Id × pMsg(P.M[P])))
⇒ add-cause(<t, x>;G) ⊆ snd(snd(let ev,x,c = lg-label(v8;v9) in
if com-kind(c) =a "msg" then <inl <ev, x, comm-msg(c)>, deliver-msg(t;comm-msg(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "create" then <inr ⋅ , create-component(t;comm-create(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "choose" then <inl <ev, x, n2m v11>, deliver-msg(t;n2m v11;x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "new" then <inl <ev, x, l2m v12>, deliver-msg(t;l2m v12;x;v7;lg-remove(v8;v9))>
else <inr ⋅ , v7, lg-remove(v8;v9)>
fi ))
BY
{ (((Assert λc.fst(c) = x ∈ component(P.M[P]) ⟶ 𝔹 BY Auto) THEN PromoteHyp (-1) 12)
THEN (GenConclAtAddrType ⌜pInTransit(P.M[P])⌝ [2;2;1;2;1;1]⋅
THENA (Auto THEN RepUR ``lg-is-source`` (-1) THEN SplitOnHypITE -1 THEN Auto)
)
THEN RepeatFor 2 (D (-2))
THEN Reduce 0
THEN RenameVar `c' (-2)
THEN Repeat (AutoSplit)
THEN (((GenConclAtAddr [2;2;1;2;2] THENA Auto) THEN Thin (-1) THEN RenameVar `mss' (-1) THEN Repeat (Thin (-2)))⋅
ORELSE (RepUR ``bfalse`` 0 THEN Auto)⋅
)
THEN Id) }
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. reliable-env(env; pRun(S;env;n2m;l2m))@i
8. ∀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)
9. t : {1...}
10. x : Id@i
11. [%12] : ↑is-run-event(pRun(S;env;n2m;l2m);t;x)@i
12. λc.fst(c) = x ∈ component(P.M[P]) ⟶ 𝔹
13. P : Process(P.M[P])@i
14. (P ∈ run-event-state-when(pRun(S;env;n2m;l2m);<t, x>))@i
15. 0 < t
16. v1 : ℤ × Id@i
17. ms : pMsg(P.M[P])@i
18. v2 : Process(P.M[P])@i
19. G : LabeledDAG(Id × pCom(P.M[P]))@i
20. Process-apply(P;ms) = <v2, G> ∈ (Process(P.M[P]) × pExt(P.M[P]))@i
21. n : ℕlg-size(G)@i
22. v3 : Id@i
23. v4 : pCom(P.M[P])@i
24. lg-label(G;n) = <v3, v4> ∈ (Id × pCom(P.M[P]))@i
25. (com-kind(v4) ∈ ``msg choose new``)@i
26. v5 : ℤ × Id × Id × pMsg(P.M[P])?@i
27. v7 : component(P.M[P]) List@i
28. v8 : LabeledDAG(pInTransit(P.M[P]))@i
29. (pRun(S;env;n2m;l2m) (t - 1)) = <v5, v7, v8> ∈ (ℤ × Id × Id × pMsg(P.M[P])? × System(P.M[P]))@i
30. v9 : ℕ@i
31. v11 : ℕ@i
32. v12 : Id@i
33. (env t pRun(S;env;n2m;l2m)) = <v9, v11, v12> ∈ (ℕ × ℕ × Id)@i
34. ↑lg-is-source(v8;v9)
35. v13 : ℤ × Id@i
36. v15 : Id@i
37. mss : pMsg(P.M[P])@i
⊢ (P ∈ mapfilter(λc.(snd(c));λc.fst(c) = x;v7))
⇒ (↑x = v15)
⇒ (<v13, mss> = <v1, ms> ∈ (ℤ × Id × pMsg(P.M[P])))
⇒ add-cause(<t, x>;G) ⊆ snd(deliver-msg(t;mss;v15;v7;lg-remove(v8;v9)))
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. reliable-env(env; pRun(S;env;n2m;l2m))@i
8. \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)))))
9. t : \{1...\}
10. x : Id@i
11. [\%12] : \muparrow{}is-run-event(pRun(S;env;n2m;l2m);t;x)@i
12. P : Process(P.M[P])@i
13. (P \mmember{} run-event-state-when(pRun(S;env;n2m;l2m);<t, x>))@i
14. 0 < t
15. v1 : \mBbbZ{} \mtimes{} Id@i
16. ms : pMsg(P.M[P])@i
17. v2 : Process(P.M[P])@i
18. G : LabeledDAG(Id \mtimes{} pCom(P.M[P]))@i
19. Process-apply(P;ms) = <v2, G>@i
20. n : \mBbbN{}lg-size(G)@i
21. v3 : Id@i
22. v4 : pCom(P.M[P])@i
23. lg-label(G;n) = <v3, v4>@i
24. (com-kind(v4) \mmember{} ``msg choose new``)@i
25. v5 : \mBbbZ{} \mtimes{} Id \mtimes{} Id \mtimes{} pMsg(P.M[P])?@i
26. v7 : component(P.M[P]) List@i
27. v8 : LabeledDAG(pInTransit(P.M[P]))@i
28. (pRun(S;env;n2m;l2m) (t - 1)) = <v5, v7, v8>@i
29. v9 : \mBbbN{}@i
30. v11 : \mBbbN{}@i
31. v12 : Id@i
32. (env t pRun(S;env;n2m;l2m)) = <v9, v11, v12>@i
33. \muparrow{}lg-is-source(v8;v9)
\mvdash{} (P \mmember{} mapfilter(\mlambda{}c.(snd(c));\mlambda{}c.fst(c) = x;v7))
{}\mRightarrow{} (\muparrow{}let info,S = let ev,x,c = lg-label(v8;v9) in
if com-kind(c) =a "msg"
then <inl <ev, x, comm-msg(c)>, deliver-msg(t;comm-msg(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "create"
then <inr \mcdot{} , create-component(t;comm-create(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "choose"
then <inl <ev, x, n2m v11>, deliver-msg(t;n2m v11;x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "new" then <inl <ev, x, l2m v12>, deliver-msg(t;l2m v12;x;v7;lg-remove(v8;v9)\000C)>
else <inr \mcdot{} , v7, lg-remove(v8;v9)>
fi
in isl(info) \mwedge{}\msubb{} let ev,z,m = outl(info) in x = z)
{}\mRightarrow{} (let info,S = let ev,x,c = lg-label(v8;v9) in
if com-kind(c) =a "msg"
then <inl <ev, x, comm-msg(c)>, deliver-msg(t;comm-msg(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "create"
then <inr \mcdot{} , create-component(t;comm-create(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "choose" then <inl <ev, x, n2m v11>, deliver-msg(t;n2m v11;x;v7;lg-remove(v8;v\000C9))>
if com-kind(c) =a "new" then <inl <ev, x, l2m v12>, deliver-msg(t;l2m v12;x;v7;lg-remove(v8;v9))\000C>
else <inr \mcdot{} , v7, lg-remove(v8;v9)>
fi
in let ev,z,m = outl(info) in
<ev, m>
= <v1, ms>)
{}\mRightarrow{} add-cause(<t, x>G) \msubseteq{} snd(snd(let ev,x,c = lg-label(v8;v9) in
if com-kind(c) =a "msg"
then <inl <ev, x, comm-msg(c)>, deliver-msg(t;comm-msg(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "create"
then <inr \mcdot{} , create-component(t;comm-create(c);x;v7;lg-remove(v8;v9))>
if com-kind(c) =a "choose" then <inl <ev, x, n2m v11>, deliver-msg(t;n2m v11;x;v7;lg-remove(v8;v9\000C))>
if com-kind(c) =a "new" then <inl <ev, x, l2m v12>, deliver-msg(t;l2m v12;x;v7;lg-remove(v8;v9))>
else <inr \mcdot{} , v7, lg-remove(v8;v9)>
fi ))
By
Latex:
(((Assert \mlambda{}c.fst(c) = x \mmember{} component(P.M[P]) {}\mrightarrow{} \mBbbB{} BY Auto) THEN PromoteHyp (-1) 12)
THEN (GenConclAtAddrType \mkleeneopen{}pInTransit(P.M[P])\mkleeneclose{} [2;2;1;2;1;1]\mcdot{}
THENA (Auto THEN RepUR ``lg-is-source`` (-1) THEN SplitOnHypITE -1 THEN Auto)
)
THEN RepeatFor 2 (D (-2))
THEN Reduce 0
THEN RenameVar `c' (-2)
THEN Repeat (AutoSplit)
THEN (((GenConclAtAddr [2;2;1;2;2] THENA Auto)
THEN Thin (-1)
THEN RenameVar `mss' (-1)
THEN Repeat (Thin (-2)))\mcdot{}
ORELSE (RepUR ``bfalse`` 0 THEN Auto)\mcdot{}
)
THEN Id)
Home
Index