Termination Proof using AProVE

Term Rewriting System R:
[X, XS, N, X1, X2]
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

2ND(cons(X, XS)) -> HEAD(activate(XS))
2ND(cons(X, XS)) -> ACTIVATE(XS)
TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_take}(X1, X2)) -> TAKE(X1, X2)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

Dependency Pairs:

TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(n_{_take}(X1, X2)) -> TAKE(X1, X2)

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

TAKE(s(N), cons(X, XS)) -> ACTIVATE(XS)

one new Dependency Pair is created:

TAKE(s(N), cons(X, n_{_take}(X1'', X2''))) -> ACTIVATE(n_{_take}(X1'', X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

Dependency Pairs:

TAKE(s(N), cons(X, n_{_take}(X1'', X2''))) -> ACTIVATE(n_{_take}(X1'', X2''))
ACTIVATE(n_{_take}(X1, X2)) -> TAKE(X1, X2)

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(n_{_take}(X1, X2)) -> TAKE(X1, X2)

one new Dependency Pair is created:

ACTIVATE(n_{_take}(s(N''), cons(X'', n_{_take}(X1'''', X2'''')))) -> TAKE(s(N''), cons(X'', n_{_take}(X1'''', X2'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳FwdInst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_take}(s(N''), cons(X'', n_{_take}(X1'''', X2'''')))) -> TAKE(s(N''), cons(X'', n_{_take}(X1'''', X2'''')))
TAKE(s(N), cons(X, n_{_take}(X1'', X2''))) -> ACTIVATE(n_{_take}(X1'', X2''))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

TAKE(s(N), cons(X, n_{_take}(X1'', X2''))) -> ACTIVATE(n_{_take}(X1'', X2''))

one new Dependency Pair is created:

TAKE(s(N), cons(X, n_{_take}(s(N''''), cons(X'''', n_{_take}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_take}(s(N''''), cons(X'''', n_{_take}(X1'''''', X2''''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳FwdInst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

Dependency Pairs:

TAKE(s(N), cons(X, n_{_take}(s(N''''), cons(X'''', n_{_take}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_take}(s(N''''), cons(X'''', n_{_take}(X1'''''', X2''''''))))
ACTIVATE(n_{_take}(s(N''), cons(X'', n_{_take}(X1'''', X2'''')))) -> TAKE(s(N''), cons(X'', n_{_take}(X1'''', X2'''')))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

ACTIVATE(n_{_take}(s(N''), cons(X'', n_{_take}(X1'''', X2'''')))) -> TAKE(s(N''), cons(X'', n_{_take}(X1'''', X2'''')))

one new Dependency Pair is created:

ACTIVATE(n_{_take}(s(N'''), cons(X''', n_{_take}(s(N''''''), cons(X'''''', n_{_take}(X1'''''''', X2'''''''')))))) -> TAKE(s(N'''), cons(X''', n_{_take}(s(N''''''), cons(X'''''', n_{_take}(X1'''''''', X2'''''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳FwdInst

...

               →DP Problem 6

                 ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_take}(s(N'''), cons(X''', n_{_take}(s(N''''''), cons(X'''''', n_{_take}(X1'''''''', X2'''''''')))))) -> TAKE(s(N'''), cons(X''', n_{_take}(s(N''''''), cons(X'''''', n_{_take}(X1'''''''', X2'''''''')))))
TAKE(s(N), cons(X, n_{_take}(s(N''''), cons(X'''', n_{_take}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_take}(s(N''''), cons(X'''', n_{_take}(X1'''''', X2''''''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_take}(s(N'''), cons(X''', n_{_take}(s(N''''''), cons(X'''''', n_{_take}(X1'''''''', X2'''''''')))))) -> TAKE(s(N'''), cons(X''', n_{_take}(s(N''''''), cons(X'''''', n_{_take}(X1'''''''', X2'''''''')))))
TAKE(s(N), cons(X, n_{_take}(s(N''''), cons(X'''', n_{_take}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_take}(s(N''''), cons(X'''', n_{_take}(X1'''''', X2''''''))))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

TAKE(x₁, x₂) -> x₂
cons(x₁, x₂) -> cons(x₁, x₂)
ACTIVATE(x₁) -> x₁
n_{_take}(x₁, x₂) -> n_{_take}(x₁, x₂)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 3

             ↳FwdInst

...

               →DP Problem 7

                 ↳Dependency Graph

       →DP Problem 2

         ↳Nar

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Narrowing Transformation

Dependency Pair:

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

three new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_from}(X''))) -> SEL(N, from(X''))
SEL(s(N), cons(X, n_{_take}(X1', X2'))) -> SEL(N, take(X1', X2'))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Narrowing Transformation

Dependency Pairs:

SEL(s(N), cons(X, XS')) -> SEL(N, XS')
SEL(s(N), cons(X, n_{_take}(X1', X2'))) -> SEL(N, take(X1', X2'))
SEL(s(N), cons(X, n_{_from}(X''))) -> SEL(N, from(X''))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, n_{_from}(X''))) -> SEL(N, from(X''))

two new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, cons(X''', n_{_from}(s(X'''))))
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, n_{_from}(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 9

                 ↳Narrowing Transformation

Dependency Pairs:

SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, cons(X''', n_{_from}(s(X'''))))
SEL(s(N), cons(X, n_{_take}(X1', X2'))) -> SEL(N, take(X1', X2'))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, n_{_take}(X1', X2'))) -> SEL(N, take(X1', X2'))

three new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_take}(0, X2''))) -> SEL(N, nil)
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', XS')))) -> SEL(N, cons(X'', n_{_take}(N'', activate(XS'))))
SEL(s(N), cons(X, n_{_take}(X1'', X2''))) -> SEL(N, n_{_take}(X1'', X2''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

Dependency Pairs:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', XS')))) -> SEL(N, cons(X'', n_{_take}(N'', activate(XS'))))
SEL(s(N), cons(X, XS')) -> SEL(N, XS')
SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, cons(X''', n_{_from}(s(X'''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, XS')) -> SEL(N, XS')

three new Dependency Pairs are created:

SEL(s(s(N'')), cons(X, cons(X'', XS'''))) -> SEL(s(N''), cons(X'', XS'''))
SEL(s(s(N'')), cons(X, cons(X'', n_{_from}(X''''')))) -> SEL(s(N''), cons(X'', n_{_from}(X''''')))
SEL(s(s(N'')), cons(X, cons(X'', n_{_take}(s(N''''), cons(X'''', XS'''))))) -> SEL(s(N''), cons(X'', n_{_take}(s(N''''), cons(X'''', XS'''))))

The transformation is resulting in three new DP problems:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 11

                 ↳Narrowing Transformation

Dependency Pair:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', XS')))) -> SEL(N, cons(X'', n_{_take}(N'', activate(XS'))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', XS')))) -> SEL(N, cons(X'', n_{_take}(N'', activate(XS'))))

three new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_from}(X'''))))) -> SEL(N, cons(X'', n_{_take}(N'', from(X'''))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(X1', X2'))))) -> SEL(N, cons(X'', n_{_take}(N'', take(X1', X2'))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', XS'')))) -> SEL(N, cons(X'', n_{_take}(N'', XS'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 16

                 ↳Narrowing Transformation

Dependency Pairs:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', XS'')))) -> SEL(N, cons(X'', n_{_take}(N'', XS'')))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(X1', X2'))))) -> SEL(N, cons(X'', n_{_take}(N'', take(X1', X2'))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_from}(X'''))))) -> SEL(N, cons(X'', n_{_take}(N'', from(X'''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_from}(X'''))))) -> SEL(N, cons(X'', n_{_take}(N'', from(X'''))))

two new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_from}(X''''))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X'''', n_{_from}(s(X''''))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_from}(X''''))))) -> SEL(N, cons(X'', n_{_take}(N'', n_{_from}(X''''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 17

                 ↳Narrowing Transformation

Dependency Pairs:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_from}(X''''))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X'''', n_{_from}(s(X''''))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(X1', X2'))))) -> SEL(N, cons(X'', n_{_take}(N'', take(X1', X2'))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', XS'')))) -> SEL(N, cons(X'', n_{_take}(N'', XS'')))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(X1', X2'))))) -> SEL(N, cons(X'', n_{_take}(N'', take(X1', X2'))))

three new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(0, X2''))))) -> SEL(N, cons(X'', n_{_take}(N'', nil)))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', XS')))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', activate(XS'))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(X1'', X2''))))) -> SEL(N, cons(X'', n_{_take}(N'', n_{_take}(X1'', X2''))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 18

                 ↳Narrowing Transformation

Dependency Pairs:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', XS')))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', activate(XS'))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', XS'')))) -> SEL(N, cons(X'', n_{_take}(N'', XS'')))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_from}(X''''))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X'''', n_{_from}(s(X''''))))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', XS')))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', activate(XS'))))))

three new Dependency Pairs are created:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', n_{_from}(X'0))))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', from(X'0))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', n_{_take}(X1', X2'))))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', take(X1', X2'))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', XS'')))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', XS'')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 19

                 ↳Forward Instantiation Transformation

Dependency Pairs:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', XS'')))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', XS'')))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', n_{_take}(X1', X2'))))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', take(X1', X2'))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', n_{_from}(X'0))))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', from(X'0))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_from}(X''''))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X'''', n_{_from}(s(X''''))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', XS'')))) -> SEL(N, cons(X'', n_{_take}(N'', XS'')))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', XS'')))) -> SEL(N, cons(X'', n_{_take}(N'', XS'')))

five new Dependency Pairs are created:

SEL(s(s(N'0)), cons(X, n_{_take}(s(s(N'''')), cons(X'''', cons(X''''', XS''''))))) -> SEL(s(N'0), cons(X'''', n_{_take}(s(N''''), cons(X''''', XS''''))))
SEL(s(s(N'0)), cons(X, n_{_take}(s(s(N'''')), cons(X'''', cons(X'''0, n_{_from}(X'''''')))))) -> SEL(s(N'0), cons(X'''', n_{_take}(s(N''''), cons(X'''0, n_{_from}(X'''''')))))
SEL(s(s(N''')), cons(X, n_{_take}(s(s(N'''''')), cons(X''', cons(X''''0, n_{_take}(s(N'''''''), cons(X'''''', n_{_from}(X'0'')))))))) -> SEL(s(N'''), cons(X''', n_{_take}(s(N''''''), cons(X''''0, n_{_take}(s(N'''''''), cons(X'''''', n_{_from}(X'0'')))))))
SEL(s(s(N''')), cons(X, n_{_take}(s(s(N'''''')), cons(X''', cons(X''''0, n_{_take}(s(N'''''''), cons(X'''''', n_{_take}(X1''', X2''')))))))) -> SEL(s(N'''), cons(X''', n_{_take}(s(N''''''), cons(X''''0, n_{_take}(s(N'''''''), cons(X'''''', n_{_take}(X1''', X2''')))))))
SEL(s(s(N''')), cons(X, n_{_take}(s(s(N'''''')), cons(X''', cons(X''''0, n_{_take}(s(N'''''''), cons(X'''''', XS''''))))))) -> SEL(s(N'''), cons(X''', n_{_take}(s(N''''''), cons(X''''0, n_{_take}(s(N'''''''), cons(X'''''', XS''''))))))

The transformation is resulting in three new DP problems:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 20

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', n_{_take}(X1', X2'))))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', take(X1', X2'))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', n_{_from}(X'0))))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', from(X'0))))))
SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_take}(s(N'''), cons(X''', XS'')))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X''', n_{_take}(N''', XS'')))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 21

                 ↳Instantiation Transformation

Dependency Pair:

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_from}(X''''))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X'''', n_{_from}(s(X''''))))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(N), cons(X, n_{_take}(s(N''), cons(X'', n_{_from}(X''''))))) -> SEL(N, cons(X'', n_{_take}(N'', cons(X'''', n_{_from}(s(X''''))))))

one new Dependency Pair is created:

SEL(s(N0), cons(X', n_{_take}(s(N''''), cons(X''0, n_{_from}(s(X''0)))))) -> SEL(N0, cons(X''0, n_{_take}(N'''', cons(s(X''0), n_{_from}(s(s(X''0)))))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 23

                 ↳Argument Filtering and Ordering

Dependency Pair:

SEL(s(N0), cons(X', n_{_take}(s(N''''), cons(X''0, n_{_from}(s(X''0)))))) -> SEL(N0, cons(X''0, n_{_take}(N'''', cons(s(X''0), n_{_from}(s(s(X''0)))))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

SEL(s(N0), cons(X', n_{_take}(s(N''''), cons(X''0, n_{_from}(s(X''0)))))) -> SEL(N0, cons(X''0, n_{_take}(N'''', cons(s(X''0), n_{_from}(s(s(X''0)))))))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

SEL(x₁, x₂) -> SEL(x₁, x₂)
s(x₁) -> s(x₁)
cons(x₁, x₂) -> x₂
n_{_take}(x₁, x₂) -> n_{_take}(x₁, x₂)
n_{_from}(x₁) -> n_{_from}

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 25

                 ↳Dependency Graph

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 22

                 ↳Forward Instantiation Transformation

Dependency Pair:

SEL(s(s(N'0)), cons(X, n_{_take}(s(s(N'''')), cons(X'''', cons(X''''', XS''''))))) -> SEL(s(N'0), cons(X'''', n_{_take}(s(N''''), cons(X''''', XS''''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

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

SEL(s(s(N'0)), cons(X, n_{_take}(s(s(N'''')), cons(X'''', cons(X''''', XS''''))))) -> SEL(s(N'0), cons(X'''', n_{_take}(s(N''''), cons(X''''', XS''''))))

one new Dependency Pair is created:

SEL(s(s(s(N'0''))), cons(X, n_{_take}(s(s(s(N''''''))), cons(X''''0, cons(X'''''0, cons(X'''''''', XS'''''')))))) -> SEL(s(s(N'0'')), cons(X''''0, n_{_take}(s(s(N'''''')), cons(X'''''0, cons(X'''''''', XS'''''')))))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 24

                 ↳Argument Filtering and Ordering

Dependency Pair:

SEL(s(s(s(N'0''))), cons(X, n_{_take}(s(s(s(N''''''))), cons(X''''0, cons(X'''''0, cons(X'''''''', XS'''''')))))) -> SEL(s(s(N'0'')), cons(X''''0, n_{_take}(s(s(N'''''')), cons(X'''''0, cons(X'''''''', XS'''''')))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

SEL(s(s(s(N'0''))), cons(X, n_{_take}(s(s(s(N''''''))), cons(X''''0, cons(X'''''0, cons(X'''''''', XS'''''')))))) -> SEL(s(s(N'0'')), cons(X''''0, n_{_take}(s(s(N'''''')), cons(X'''''0, cons(X'''''''', XS'''''')))))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

SEL(x₁, x₂) -> SEL(x₁, x₂)
s(x₁) -> s(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)
n_{_take}(x₁, x₂) -> x₂

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 12

                 ↳Argument Filtering and Ordering

Dependency Pair:

SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, cons(X''', n_{_from}(s(X'''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

SEL(s(N), cons(X, n_{_from}(X'''))) -> SEL(N, cons(X''', n_{_from}(s(X'''))))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

SEL(x₁, x₂) -> SEL(x₁, x₂)
s(x₁) -> s(x₁)
cons(x₁, x₂) -> x₂
n_{_from}(x₁) -> n_{_from}

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 14

                 ↳Dependency Graph

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Nar

           →DP Problem 8

             ↳Nar

...

               →DP Problem 13

                 ↳Argument Filtering and Ordering

Dependency Pair:

SEL(s(s(N'')), cons(X, cons(X'', XS'''))) -> SEL(s(N''), cons(X'', XS'''))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
head(cons(X, XS)) -> X
2nd(cons(X, XS)) -> head(activate(XS))
take(0, XS) -> nil
take(s(N), cons(X, XS)) -> cons(X, n_{_take}(N, activate(XS)))
take(X1, X2) -> n_{_take}(X1, X2)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(n_{_from}(X)) -> from(X)
activate(n_{_take}(X1, X2)) -> take(X1, X2)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

SEL(s(s(N'')), cons(X, cons(X'', XS'''))) -> SEL(s(N''), cons(X'', XS'''))

There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:

SEL(x₁, x₂) -> SEL(x₁, x₂)
s(x₁) -> s(x₁)
cons(x₁, x₂) -> cons(x₁, x₂)

Innermost Termination of R could not be shown.
Duration:
0:08 minutes