Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z]
f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

G(s(X)) -> G(X)
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_f}(X)) -> F(activate(X))
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_g}(X)) -> G(activate(X))
ACTIVATE(n_{_g}(X)) -> ACTIVATE(X)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

Dependency Pair:

G(s(X)) -> G(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

The following dependency pair can be strictly oriented:

G(s(X)) -> G(X)

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

G(x₁) -> G(x₁)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 4

             ↳Dependency Graph

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Nar

Dependency Pairs:

ACTIVATE(n_{_g}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

The following dependency pairs can be strictly oriented:

ACTIVATE(n_{_g}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)

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

ACTIVATE(x₁) -> ACTIVATE(x₁)
n_{_g}(x₁) -> n_{_g}(x₁)
n_{_f}(x₁) -> n_{_f}(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

           →DP Problem 5

             ↳Dependency Graph

       →DP Problem 3

         ↳Nar

Dependency Pair:

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Narrowing Transformation

Dependency Pair:

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

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

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_{_f}(X''))) -> SEL(X, f(activate(X'')))
SEL(s(X), cons(Y, n_{_g}(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Narrowing Transformation

Dependency Pairs:

SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, n_{_g}(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, n_{_f}(X''))) -> SEL(X, f(activate(X'')))

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

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

SEL(s(X), cons(Y, n_{_f}(X''))) -> SEL(X, f(activate(X'')))

five new Dependency Pairs are created:

SEL(s(X), cons(Y, n_{_f}(X'''))) -> SEL(X, cons(activate(X'''), n_{_f}(n_{_g}(activate(X''')))))
SEL(s(X), cons(Y, n_{_f}(X'''))) -> SEL(X, n_{_f}(activate(X''')))
SEL(s(X), cons(Y, n_{_f}(n_{_f}(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, n_{_f}(n_{_g}(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, n_{_f}(X'''))) -> SEL(X, f(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

Dependency Pairs:

SEL(s(X), cons(Y, n_{_f}(X'''))) -> SEL(X, f(X'''))
SEL(s(X), cons(Y, n_{_f}(n_{_f}(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, n_{_f}(n_{_g}(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, n_{_f}(X'''))) -> SEL(X, cons(activate(X'''), n_{_f}(n_{_g}(activate(X''')))))
SEL(s(X), cons(Y, n_{_g}(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

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

SEL(s(X), cons(Y, n_{_g}(X''))) -> SEL(X, g(activate(X'')))

four new Dependency Pairs are created:

SEL(s(X), cons(Y, n_{_g}(X'''))) -> SEL(X, n_{_g}(activate(X''')))
SEL(s(X), cons(Y, n_{_g}(n_{_f}(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(X), cons(Y, n_{_g}(n_{_g}(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, n_{_g}(X'''))) -> SEL(X, g(X'''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 8

                 ↳Argument Filtering and Ordering

Dependency Pairs:

SEL(s(X), cons(Y, n_{_g}(X'''))) -> SEL(X, g(X'''))
SEL(s(X), cons(Y, n_{_g}(n_{_g}(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, n_{_g}(n_{_f}(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(X), cons(Y, n_{_f}(n_{_f}(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, n_{_f}(n_{_g}(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, n_{_f}(X'''))) -> SEL(X, cons(activate(X'''), n_{_f}(n_{_g}(activate(X''')))))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, n_{_f}(X'''))) -> SEL(X, f(X'''))

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

The following dependency pairs can be strictly oriented:

SEL(s(X), cons(Y, n_{_g}(X'''))) -> SEL(X, g(X'''))
SEL(s(X), cons(Y, n_{_g}(n_{_g}(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, n_{_g}(n_{_f}(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(X), cons(Y, n_{_f}(n_{_f}(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, n_{_f}(n_{_g}(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, n_{_f}(X'''))) -> SEL(X, cons(activate(X'''), n_{_f}(n_{_g}(activate(X''')))))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, n_{_f}(X'''))) -> SEL(X, f(X'''))

There are no usable rules w.r.t. to the AFS 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) -> x₁
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

       →DP Problem 2

         ↳AFS

       →DP Problem 3

         ↳Nar

           →DP Problem 6

             ↳Nar

...

               →DP Problem 9

                 ↳Dependency Graph

Dependency Pair:

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes