Term Rewriting System R:
[X, XS, N]
from(X) -> cons(X, from(s(X)))
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(XS)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, XS)

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

FROM(X) -> FROM(s(X))
2ND(cons(X, XS)) -> HEAD(XS)
TAKE(s(N), cons(X, XS)) -> TAKE(N, XS)
SEL(s(N), cons(X, XS)) -> SEL(N, XS)

Furthermore, R contains three SCCs.


   R
DPs
       →DP Problem 1
Instantiation Transformation
       →DP Problem 2
Remaining
       →DP Problem 3
Remaining


Dependency Pair:

FROM(X) -> FROM(s(X))


Rules:


from(X) -> cons(X, from(s(X)))
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(XS)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, XS)


Strategy:

innermost




On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

FROM(X) -> FROM(s(X))
one new Dependency Pair is created:

FROM(s(X'')) -> FROM(s(s(X'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Inst
           →DP Problem 4
Instantiation Transformation
       →DP Problem 2
Remaining
       →DP Problem 3
Remaining


Dependency Pair:

FROM(s(X'')) -> FROM(s(s(X'')))


Rules:


from(X) -> cons(X, from(s(X)))
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(XS)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, XS)


Strategy:

innermost




On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

FROM(s(X'')) -> FROM(s(s(X'')))
one new Dependency Pair is created:

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Inst
           →DP Problem 4
Inst
             ...
               →DP Problem 5
Instantiation Transformation
       →DP Problem 2
Remaining
       →DP Problem 3
Remaining


Dependency Pair:

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))


Rules:


from(X) -> cons(X, from(s(X)))
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(XS)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, XS)


Strategy:

innermost




On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

FROM(s(s(X''''))) -> FROM(s(s(s(X''''))))
one new Dependency Pair is created:

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Inst
           →DP Problem 4
Inst
             ...
               →DP Problem 6
Instantiation Transformation
       →DP Problem 2
Remaining
       →DP Problem 3
Remaining


Dependency Pair:

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))


Rules:


from(X) -> cons(X, from(s(X)))
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(XS)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, XS)


Strategy:

innermost




On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

FROM(s(s(s(X'''''')))) -> FROM(s(s(s(s(X'''''')))))
one new Dependency Pair is created:

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Inst
           →DP Problem 4
Inst
             ...
               →DP Problem 7
Instantiation Transformation
       →DP Problem 2
Remaining
       →DP Problem 3
Remaining


Dependency Pair:

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))


Rules:


from(X) -> cons(X, from(s(X)))
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(XS)
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, take(N, XS))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, XS)


Strategy:

innermost




On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

FROM(s(s(s(s(X''''''''))))) -> FROM(s(s(s(s(s(X''''''''))))))
one new Dependency Pair is created:

FROM(s(s(s(s(s(X'''''''''')))))) -> FROM(s(s(s(s(s(s(X'''''''''')))))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Inst
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)




The following remains to be proven:


   R
DPs
       →DP Problem 1
Inst
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)




The following remains to be proven:


   R
DPs
       →DP Problem 1
Inst
       →DP Problem 2
Remaining Obligation(s)
       →DP Problem 3
Remaining Obligation(s)




The following remains to be proven:

Innermost Termination of R could not be shown.
Duration:
0:00 minutes