Termination Proof using AProVE

Term Rewriting System R:
[X, XS, N]
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

AFTER(s(N), cons(X, XS)) -> AFTER(N, activate(XS))
AFTER(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(n_{_from}(X)) -> FROM(activate(X))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_s}(X)) -> S(activate(X))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__from(x₁)) = x₁

POL(n__s(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 3

             ↳Polynomial Ordering

       →DP Problem 2

         ↳Nar

Dependency Pair:

ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__from(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 3

             ↳Polo

...

               →DP Problem 4

                 ↳Dependency Graph

       →DP Problem 2

         ↳Nar

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Narrowing Transformation

Dependency Pair:

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

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

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

three new Dependency Pairs are created:

AFTER(s(N), cons(X, n_{_from}(X''))) -> AFTER(N, from(activate(X'')))
AFTER(s(N), cons(X, n_{_s}(X''))) -> AFTER(N, s(activate(X'')))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 5

             ↳Narrowing Transformation

Dependency Pairs:

AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')
AFTER(s(N), cons(X, n_{_s}(X''))) -> AFTER(N, s(activate(X'')))
AFTER(s(N), cons(X, n_{_from}(X''))) -> AFTER(N, from(activate(X'')))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

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

five new Dependency Pairs are created:

AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, n_{_from}(activate(X''')))
AFTER(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> AFTER(N, from(from(activate(X'''))))
AFTER(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> AFTER(N, from(s(activate(X'''))))
AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, from(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 5

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pairs:

AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, from(X'''))
AFTER(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> AFTER(N, from(from(activate(X'''))))
AFTER(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> AFTER(N, from(s(activate(X'''))))
AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
AFTER(s(N), cons(X, n_{_s}(X''))) -> AFTER(N, s(activate(X'')))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

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

AFTER(s(N), cons(X, n_{_s}(X''))) -> AFTER(N, s(activate(X'')))

four new Dependency Pairs are created:

AFTER(s(N), cons(X, n_{_s}(X'''))) -> AFTER(N, n_{_s}(activate(X''')))
AFTER(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> AFTER(N, s(from(activate(X'''))))
AFTER(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> AFTER(N, s(s(activate(X'''))))
AFTER(s(N), cons(X, n_{_s}(X'''))) -> AFTER(N, s(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 5

             ↳Nar

...

               →DP Problem 7

                 ↳Polynomial Ordering

Dependency Pairs:

AFTER(s(N), cons(X, n_{_s}(X'''))) -> AFTER(N, s(X'''))
AFTER(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> AFTER(N, s(s(activate(X'''))))
AFTER(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> AFTER(N, s(from(activate(X'''))))
AFTER(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> AFTER(N, from(from(activate(X'''))))
AFTER(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> AFTER(N, from(s(activate(X'''))))
AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')
AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, from(X'''))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

The following dependency pairs can be strictly oriented:

AFTER(s(N), cons(X, n_{_s}(X'''))) -> AFTER(N, s(X'''))
AFTER(s(N), cons(X, n_{_s}(n_{_s}(X''')))) -> AFTER(N, s(s(activate(X'''))))
AFTER(s(N), cons(X, n_{_s}(n_{_from}(X''')))) -> AFTER(N, s(from(activate(X'''))))
AFTER(s(N), cons(X, n_{_from}(n_{_from}(X''')))) -> AFTER(N, from(from(activate(X'''))))
AFTER(s(N), cons(X, n_{_from}(n_{_s}(X''')))) -> AFTER(N, from(s(activate(X'''))))
AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, cons(activate(X'''), n_{_from}(n_{_s}(activate(X''')))))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')
AFTER(s(N), cons(X, n_{_from}(X'''))) -> AFTER(N, from(X'''))

Additionally, the following usable rules using the Ce-refinement can be oriented:

s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__from(x₁)) = 0

POL(from(x₁)) = 0

POL(activate(x₁)) = x₁

POL(AFTER(x₁, x₂)) = x₁

POL(cons(x₁, x₂)) = 0

POL(n__s(x₁)) = 1 + x₁

POL(s(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Nar

           →DP Problem 5

             ↳Nar

...

               →DP Problem 8

                 ↳Dependency Graph

Dependency Pair:

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:01 minutes

POL(n__from(x₁))	= x₁
POL(n__s(x₁))	= 1 + x₁
POL(ACTIVATE(x₁))	= x₁

POL(n__from(x₁))	= 0
POL(from(x₁))	= 0
POL(activate(x₁))	= x₁
POL(AFTER(x₁, x₂))	= x₁
POL(cons(x₁, x₂))	= 0
POL(n__s(x₁))	= 1 + x₁
POL(s(x₁))	= 1 + x₁