Termination Proof using AProVE

Term Rewriting System R:
[X, Z, Y, X1, X2]
from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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

FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

one new Dependency Pair is created:

FIRST(s(X), cons(Y, n_{_first}(X1'', X2''))) -> ACTIVATE(n_{_first}(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:

FIRST(s(X), cons(Y, n_{_first}(X1'', X2''))) -> ACTIVATE(n_{_first}(X1'', X2''))
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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_{_first}(X1, X2)) -> FIRST(X1, X2)

one new Dependency Pair is created:

ACTIVATE(n_{_first}(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', n_{_first}(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_{_first}(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))
FIRST(s(X), cons(Y, n_{_first}(X1'', X2''))) -> ACTIVATE(n_{_first}(X1'', X2''))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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

FIRST(s(X), cons(Y, n_{_first}(X1'', X2''))) -> ACTIVATE(n_{_first}(X1'', X2''))

one new Dependency Pair is created:

FIRST(s(X), cons(Y, n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_first}(s(X''''), cons(Y'''', n_{_first}(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:

FIRST(s(X), cons(Y, n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))
ACTIVATE(n_{_first}(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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_{_first}(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))) -> FIRST(s(X''), cons(Y'', n_{_first}(X1'''', X2'''')))

one new Dependency Pair is created:

ACTIVATE(n_{_first}(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))) -> FIRST(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(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_{_first}(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))) -> FIRST(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))
FIRST(s(X), cons(Y, n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_first}(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))) -> FIRST(s(X'''), cons(Y''', n_{_first}(s(X''''''), cons(Y'''''', n_{_first}(X1'''''''', X2'''''''')))))
FIRST(s(X), cons(Y, n_{_first}(s(X''''), cons(Y'''', n_{_first}(X1'''''', X2''''''))))) -> ACTIVATE(n_{_first}(s(X''''), cons(Y'''', n_{_first}(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:

FIRST(x₁, x₂) -> x₂
cons(x₁, x₂) -> cons(x₁, x₂)
ACTIVATE(x₁) -> x₁
n_{_first}(x₁, x₂) -> n_{_first}(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)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, Z)) -> SEL(X, activate(Z))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, Z)) -> SEL(X, activate(Z))

three new Dependency Pairs are created:

SEL(s(X), cons(Y, n_{_from}(X''))) -> SEL(X, from(X''))
SEL(s(X), cons(Y, n_{_first}(X1', X2'))) -> SEL(X, first(X1', X2'))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')

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(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, n_{_first}(X1', X2'))) -> SEL(X, first(X1', X2'))
SEL(s(X), cons(Y, n_{_from}(X''))) -> SEL(X, from(X''))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, n_{_from}(X''))) -> SEL(X, from(X''))

two new Dependency Pairs are created:

SEL(s(X), cons(Y, n_{_from}(X'''))) -> SEL(X, cons(X''', n_{_from}(s(X'''))))
SEL(s(X), cons(Y, n_{_from}(X'''))) -> SEL(X, 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(X), cons(Y, n_{_from}(X'''))) -> SEL(X, cons(X''', n_{_from}(s(X'''))))
SEL(s(X), cons(Y, n_{_first}(X1', X2'))) -> SEL(X, first(X1', X2'))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, n_{_first}(X1', X2'))) -> SEL(X, first(X1', X2'))

three new Dependency Pairs are created:

SEL(s(X), cons(Y, n_{_first}(0, X2''))) -> SEL(X, nil)
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', Z')))) -> SEL(X, cons(Y'', n_{_first}(X'', activate(Z'))))
SEL(s(X), cons(Y, n_{_first}(X1'', X2''))) -> SEL(X, n_{_first}(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(X), cons(Y, n_{_first}(s(X''), cons(Y'', Z')))) -> SEL(X, cons(Y'', n_{_first}(X'', activate(Z'))))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, n_{_from}(X'''))) -> SEL(X, cons(X''', n_{_from}(s(X'''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, Z')) -> SEL(X, Z')

three new Dependency Pairs are created:

SEL(s(s(X'')), cons(Y, cons(Y'', Z'''))) -> SEL(s(X''), cons(Y'', Z'''))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_from}(X''''')))) -> SEL(s(X''), cons(Y'', n_{_from}(X''''')))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_first}(s(X''''), cons(Y'''', Z'''))))) -> SEL(s(X''), cons(Y'', n_{_first}(s(X''''), cons(Y'''', Z'''))))

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(X), cons(Y, n_{_first}(s(X''), cons(Y'', Z')))) -> SEL(X, cons(Y'', n_{_first}(X'', activate(Z'))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, n_{_first}(s(X''), cons(Y'', Z')))) -> SEL(X, cons(Y'', n_{_first}(X'', activate(Z'))))

three new Dependency Pairs are created:

SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_from}(X'''))))) -> SEL(X, cons(Y'', n_{_first}(X'', from(X'''))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(X1', X2'))))) -> SEL(X, cons(Y'', n_{_first}(X'', first(X1', X2'))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', Z'')))) -> SEL(X, cons(Y'', n_{_first}(X'', Z'')))

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(X), cons(Y, n_{_first}(s(X''), cons(Y'', Z'')))) -> SEL(X, cons(Y'', n_{_first}(X'', Z'')))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(X1', X2'))))) -> SEL(X, cons(Y'', n_{_first}(X'', first(X1', X2'))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_from}(X'''))))) -> SEL(X, cons(Y'', n_{_first}(X'', from(X'''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_from}(X'''))))) -> SEL(X, cons(Y'', n_{_first}(X'', from(X'''))))

two new Dependency Pairs are created:

SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_from}(X''''))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(X'''', n_{_from}(s(X''''))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_from}(X''''))))) -> SEL(X, cons(Y'', n_{_first}(X'', 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(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_from}(X''''))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(X'''', n_{_from}(s(X''''))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(X1', X2'))))) -> SEL(X, cons(Y'', n_{_first}(X'', first(X1', X2'))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', Z'')))) -> SEL(X, cons(Y'', n_{_first}(X'', Z'')))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(X1', X2'))))) -> SEL(X, cons(Y'', n_{_first}(X'', first(X1', X2'))))

three new Dependency Pairs are created:

SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(0, X2''))))) -> SEL(X, cons(Y'', n_{_first}(X'', nil)))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', Z')))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', activate(Z'))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(X1'', X2''))))) -> SEL(X, cons(Y'', n_{_first}(X'', n_{_first}(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(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', Z')))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', activate(Z'))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', Z'')))) -> SEL(X, cons(Y'', n_{_first}(X'', Z'')))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_from}(X''''))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(X'''', n_{_from}(s(X''''))))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', Z')))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', activate(Z'))))))

three new Dependency Pairs are created:

SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', n_{_from}(X'0))))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', from(X'0))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', n_{_first}(X1', X2'))))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', first(X1', X2'))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', Z'')))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', Z'')))))

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(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', Z'')))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', Z'')))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', n_{_first}(X1', X2'))))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', first(X1', X2'))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', n_{_from}(X'0))))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', from(X'0))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_from}(X''''))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(X'''', n_{_from}(s(X''''))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', Z'')))) -> SEL(X, cons(Y'', n_{_first}(X'', Z'')))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, n_{_first}(s(X''), cons(Y'', Z'')))) -> SEL(X, cons(Y'', n_{_first}(X'', Z'')))

five new Dependency Pairs are created:

SEL(s(s(X'0)), cons(Y, n_{_first}(s(s(X'''')), cons(Y''0, cons(Y'''', Z''''))))) -> SEL(s(X'0), cons(Y''0, n_{_first}(s(X''''), cons(Y'''', Z''''))))
SEL(s(s(X'0)), cons(Y, n_{_first}(s(s(X'''')), cons(Y''0, cons(Y'''', n_{_from}(X'''''')))))) -> SEL(s(X'0), cons(Y''0, n_{_first}(s(X''''), cons(Y'''', n_{_from}(X'''''')))))
SEL(s(s(X''')), cons(Y, n_{_first}(s(s(X''''0)), cons(Y''', cons(Y'''''', n_{_first}(s(X''''''), cons(Y''''''', n_{_from}(X'0'')))))))) -> SEL(s(X'''), cons(Y''', n_{_first}(s(X''''0), cons(Y'''''', n_{_first}(s(X''''''), cons(Y''''''', n_{_from}(X'0'')))))))
SEL(s(s(X''')), cons(Y, n_{_first}(s(s(X''''0)), cons(Y''', cons(Y'''''', n_{_first}(s(X''''''), cons(Y''''''', n_{_first}(X1''', X2''')))))))) -> SEL(s(X'''), cons(Y''', n_{_first}(s(X''''0), cons(Y'''''', n_{_first}(s(X''''''), cons(Y''''''', n_{_first}(X1''', X2''')))))))
SEL(s(s(X''')), cons(Y, n_{_first}(s(s(X''''0)), cons(Y''', cons(Y'''''', n_{_first}(s(X''''''), cons(Y''''''', Z''''))))))) -> SEL(s(X'''), cons(Y''', n_{_first}(s(X''''0), cons(Y'''''', n_{_first}(s(X''''''), cons(Y''''''', Z''''))))))

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(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', n_{_first}(X1', X2'))))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', first(X1', X2'))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', n_{_from}(X'0))))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', from(X'0))))))
SEL(s(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_first}(s(X'''), cons(Y''', Z'')))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(Y''', n_{_first}(X''', Z'')))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_from}(X''''))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(X'''', n_{_from}(s(X''''))))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X), cons(Y, n_{_first}(s(X''), cons(Y'', n_{_from}(X''''))))) -> SEL(X, cons(Y'', n_{_first}(X'', cons(X'''', n_{_from}(s(X''''))))))

one new Dependency Pair is created:

SEL(s(X0), cons(Y', n_{_first}(s(X''0), cons(Y'''', n_{_from}(s(Y'''')))))) -> SEL(X0, cons(Y'''', n_{_first}(X''0, cons(s(Y''''), n_{_from}(s(s(Y'''')))))))

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(X0), cons(Y', n_{_first}(s(X''0), cons(Y'''', n_{_from}(s(Y'''')))))) -> SEL(X0, cons(Y'''', n_{_first}(X''0, cons(s(Y''''), n_{_from}(s(s(Y'''')))))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

SEL(s(X0), cons(Y', n_{_first}(s(X''0), cons(Y'''', n_{_from}(s(Y'''')))))) -> SEL(X0, cons(Y'''', n_{_first}(X''0, cons(s(Y''''), n_{_from}(s(s(Y'''')))))))

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_{_first}(x₁, x₂) -> n_{_first}(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)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X'0)), cons(Y, n_{_first}(s(s(X'''')), cons(Y''0, cons(Y'''', Z''''))))) -> SEL(s(X'0), cons(Y''0, n_{_first}(s(X''''), cons(Y'''', Z''''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X'0)), cons(Y, n_{_first}(s(s(X'''')), cons(Y''0, cons(Y'''', Z''''))))) -> SEL(s(X'0), cons(Y''0, n_{_first}(s(X''''), cons(Y'''', Z''''))))

one new Dependency Pair is created:

SEL(s(s(s(X'0''))), cons(Y, n_{_first}(s(s(s(X''''''))), cons(Y''0'', cons(Y'''''', cons(Y''''''', Z'''''')))))) -> SEL(s(s(X'0'')), cons(Y''0'', n_{_first}(s(s(X'''''')), cons(Y'''''', cons(Y''''''', Z'''''')))))

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(X'0''))), cons(Y, n_{_first}(s(s(s(X''''''))), cons(Y''0'', cons(Y'''''', cons(Y''''''', Z'''''')))))) -> SEL(s(s(X'0'')), cons(Y''0'', n_{_first}(s(s(X'''''')), cons(Y'''''', cons(Y''''''', Z'''''')))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

SEL(s(s(s(X'0''))), cons(Y, n_{_first}(s(s(s(X''''''))), cons(Y''0'', cons(Y'''''', cons(Y''''''', Z'''''')))))) -> SEL(s(s(X'0'')), cons(Y''0'', n_{_first}(s(s(X'''''')), cons(Y'''''', cons(Y''''''', Z'''''')))))

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_{_first}(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(X), cons(Y, n_{_from}(X'''))) -> SEL(X, cons(X''', n_{_from}(s(X'''))))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

SEL(s(X), cons(Y, n_{_from}(X'''))) -> SEL(X, 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)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(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(X'')), cons(Y, cons(Y'', Z'''))) -> SEL(s(X''), cons(Y'', Z'''))

Rules:

from(X) -> cons(X, n_{_from}(s(X)))
from(X) -> n_{_from}(X)
first(0, Z) -> nil
first(s(X), cons(Y, Z)) -> cons(Y, n_{_first}(X, activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
sel(0, cons(X, Z)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_from}(X)) -> from(X)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

SEL(s(s(X'')), cons(Y, cons(Y'', Z'''))) -> SEL(s(X''), cons(Y'', Z'''))

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:
0:06 minutes