Step
*
1
1
1
1
1
1
1
of Lemma
sequence-class-program_wf
1. Info : Type
2. A : Type
3. B : Type
4. valueall-type(A)
5. es : EO+(Info)@i'
6. e : E@i
7. xpr : hdataflow(Info;A)@i
8. ypr : hdataflow(Info;B)@i
9. zpr : hdataflow(Info;A)@i
10. x : E@i
11. x ≤loc e @i
12. ↑(bag-null(snd(xpr*(map(λx.info(x);before(x)))(info(x))))
∧b (¬bbag-null(snd(ypr*(map(λx.info(x);before(x)))(info(x))))))
13. ∀e'':E
((e'' <loc x)
⇒ (¬↑(bag-null(snd(xpr*(map(λx.info(x);before(e'')))(info(e''))))
∧b (¬bbag-null(snd(ypr*(map(λx.info(x);before(e'')))(info(e''))))))))
14. v : E List@i
15. before(x) = v ∈ (E List)@i
16. ∀u:E List. ∀y:E.
(u @ [y] ≤ v
⇒ (¬↑(bag-null(snd(xpr*(map(λx.info(x);u))(info(y)))) ∧b (¬bbag-null(snd(ypr*(map(λx.info(x);u))(info(y))))))))
⊢ mk-hdf(S,a.case S
of inl(XY) =>
let X,Y = XY
in let X',bx = X(a)
in let Y',by = Y(a)
in if bag-null(bx) ∧b (¬bbag-null(by)) then let Z',bz = zpr(a) in <inr Z' , bz> else <inl <X', Y'>, bx> fi
| inr(Z) =>
let Z',b = Z(a)
in <inr Z' , b>;S.case S of inl(XY) => ff | inr(Z) => hdf-halted(Z);inl <xpr, ypr>)*(map(λx.info(x);v))
= mk-hdf(S,a.case S
of inl(XY) =>
let X,Y = XY
in let X',bx = X(a)
in let Y',by = Y(a)
in if bag-null(bx) ∧b (¬bbag-null(by)) then let Z',bz = zpr(a) in <inr Z' , bz> else <inl <X', Y'>, bx> fi
| inr(Z) =>
let Z',b = Z(a)
in <inr Z' , b>;S.case S of inl(XY) => ff | inr(Z) => hdf-halted(Z);inl <xpr*(map(λx.info(x);v))
, ypr*(map(λx.info(x);v))
>)
∈ hdataflow(Info;A)
BY
{ (ThinVar `x'
THEN MoveToConcl (-1)
THEN RepeatFor 3 (MoveToConcl (-2))
THEN ListInd (-1)
THEN Reduce 0
THEN (UnivCD THENA Auto)
THEN Try (Complete ((Fold `member` 0
THEN Using [`S',⌜hdataflow(Info;A) × hdataflow(Info;B) + hdataflow(Info;A)⌝] MemCD⋅
THEN Auto)))) }
1
1. Info : Type
2. A : Type
3. B : Type
4. valueall-type(A)
5. es : EO+(Info)@i'
6. e : E@i
7. u : E
8. v : E List
9. ∀xpr:hdataflow(Info;A). ∀ypr:hdataflow(Info;B). ∀zpr:hdataflow(Info;A).
((∀u:E List. ∀y:E.
(u @ [y] ≤ v
⇒ (¬↑(bag-null(snd(xpr*(map(λx.info(x);u))(info(y))))
∧b (¬bbag-null(snd(ypr*(map(λx.info(x);u))(info(y)))))))))
⇒ (mk-hdf(S,a.case S
of inl(XY) =>
let X,Y = XY
in let X',bx = X(a)
in let Y',by = Y(a)
in if bag-null(bx) ∧b (¬bbag-null(by)) then let Z',bz = zpr(a) in <inr Z' , bz> else <inl <X', Y'>, bx> f\000Ci
| inr(Z) =>
let Z',b = Z(a)
in <inr Z' , b>;S.case S of inl(XY) => ff | inr(Z) => hdf-halted(Z);inl <xpr, ypr>)*(map(λx.info(x);v))
= mk-hdf(S,a.case S
of inl(XY) =>
let X,Y = XY
in let X',bx = X(a)
in let Y',by = Y(a)
in if bag-null(bx) ∧b (¬bbag-null(by)) then let Z',bz = zpr(a) in <inr Z' , bz> else <inl <X', Y'>, bx>\000C fi
| inr(Z) =>
let Z',b = Z(a)
in <inr Z' , b>;S.case S of inl(XY) => ff | inr(Z) => hdf-halted(Z);inl <xpr*(map(λx.info(x);v))
, ypr*(map(λx.info(x);v))
>)
∈ hdataflow(Info;A)))
10. xpr : hdataflow(Info;A)@i
11. ypr : hdataflow(Info;B)@i
12. zpr : hdataflow(Info;A)@i
13. ∀u@0:E List. ∀y:E.
(u@0 @ [y] ≤ [u / v]
⇒ (¬↑(bag-null(snd(xpr*(map(λx.info(x);u@0))(info(y))))
∧b (¬bbag-null(snd(ypr*(map(λx.info(x);u@0))(info(y))))))))@i
⊢ fst(mk-hdf(S,a.case S
of inl(XY) =>
let X,Y = XY
in let X',bx = X(a)
in let Y',by = Y(a)
in if bag-null(bx) ∧b (¬bbag-null(by)) then let Z',bz = zpr(a) in <inr Z' , bz> else <inl <X', Y'>, bx> fi
| inr(Z) =>
let Z',b = Z(a)
in <inr Z' , b>;S.case S of inl(XY) => ff | inr(Z) => hdf-halted(Z);inl <xpr, ypr>)(info(u)))*(map(λx.info(x);v))
= mk-hdf(S,a.case S
of inl(XY) =>
let X,Y = XY
in let X',bx = X(a)
in let Y',by = Y(a)
in if bag-null(bx) ∧b (¬bbag-null(by)) then let Z',bz = zpr(a) in <inr Z' , bz> else <inl <X', Y'>, bx> fi
| inr(Z) =>
let Z',b = Z(a)
in <inr Z' , b>;S.case S of inl(XY) => ff | inr(Z) => hdf-halted(Z);inl <fst(xpr(info(u)))*(map(λx.info(x);v))
, fst(ypr(info(u)))*(map(λx.info(x);v))
>)
∈ hdataflow(Info;A)
Latex:
Latex:
1. Info : Type
2. A : Type
3. B : Type
4. valueall-type(A)
5. es : EO+(Info)@i'
6. e : E@i
7. xpr : hdataflow(Info;A)@i
8. ypr : hdataflow(Info;B)@i
9. zpr : hdataflow(Info;A)@i
10. x : E@i
11. x \mleq{}loc e @i
12. \muparrow{}(bag-null(snd(xpr*(map(\mlambda{}x.info(x);before(x)))(info(x))))
\mwedge{}\msubb{} (\mneg{}\msubb{}bag-null(snd(ypr*(map(\mlambda{}x.info(x);before(x)))(info(x))))))
13. \mforall{}e'':E
((e'' <loc x)
{}\mRightarrow{} (\mneg{}\muparrow{}(bag-null(snd(xpr*(map(\mlambda{}x.info(x);before(e'')))(info(e''))))
\mwedge{}\msubb{} (\mneg{}\msubb{}bag-null(snd(ypr*(map(\mlambda{}x.info(x);before(e'')))(info(e''))))))))
14. v : E List@i
15. before(x) = v@i
16. \mforall{}u:E List. \mforall{}y:E.
(u @ [y] \mleq{} v
{}\mRightarrow{} (\mneg{}\muparrow{}(bag-null(snd(xpr*(map(\mlambda{}x.info(x);u))(info(y))))
\mwedge{}\msubb{} (\mneg{}\msubb{}bag-null(snd(ypr*(map(\mlambda{}x.info(x);u))(info(y))))))))
\mvdash{} mk-hdf(S,a.case S
of inl(XY) =>
let X,Y = XY
in let X',bx = X(a)
in let Y',by = Y(a)
in if bag-null(bx) \mwedge{}\msubb{} (\mneg{}\msubb{}bag-null(by))
then let Z',bz = zpr(a)
in <inr Z' , bz>
else <inl <X', Y'>, bx>
fi
| inr(Z) =>
let Z',b = Z(a)
in <inr Z' , b>S.case S of inl(XY) => ff | inr(Z) => hdf-halted(Z);inl <xpr, ypr>)*(map(\mlambda{}x.info(x)\000C;v))
= mk-hdf(S,a.case S
of inl(XY) =>
let X,Y = XY
in let X',bx = X(a)
in let Y',by = Y(a)
in if bag-null(bx) \mwedge{}\msubb{} (\mneg{}\msubb{}bag-null(by))
then let Z',bz = zpr(a)
in <inr Z' , bz>
else <inl <X', Y'>, bx>
fi
| inr(Z) =>
let Z',b = Z(a)
in <inr Z' , b>S.case S of inl(XY) => ff | inr(Z) => hdf-halted(Z);inl <xpr*(map(\mlambda{}x.info(x);v))
, ypr*(map(\mlambda{}x.info(x);v))
>)
By
Latex:
(ThinVar `x'
THEN MoveToConcl (-1)
THEN RepeatFor 3 (MoveToConcl (-2))
THEN ListInd (-1)
THEN Reduce 0
THEN (UnivCD THENA Auto)
THEN Try (Complete ((Fold `member` 0
THEN Using [`S',\mkleeneopen{}hdataflow(Info;A) \mtimes{} hdataflow(Info;B) + hdataflow(Info;A)\mkleeneclose{}
] MemCD\mcdot{}
THEN Auto))))
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