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

Innermost 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

         ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →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

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳Forward Instantiation Transformation

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

Dependency Pair:

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

Strategy:

innermost

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 5

                 ↳Argument Filtering and Ordering

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

Dependency Pair:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

G(x₁) -> G(x₁)
s(x₁) -> s(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 4

             ↳FwdInst

...

               →DP Problem 6

                 ↳Dependency Graph

       →DP Problem 2

         ↳FwdInst

       →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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳Forward Instantiation Transformation

       →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

Strategy:

innermost

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Nar

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Nar

Dependency Pairs:

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

Strategy:

innermost

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 9

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Nar

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Nar

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 11

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Nar

Dependency Pairs:

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

Strategy:

innermost

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

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

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 12

                 ↳Forward Instantiation Transformation

       →DP Problem 3

         ↳Nar

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 13

                 ↳Argument Filtering and Ordering

       →DP Problem 3

         ↳Nar

Dependency Pairs:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

trivial

resulting in one new DP problem.
Used Argument Filtering System:

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

           →DP Problem 7

             ↳FwdInst

...

               →DP Problem 14

                 ↳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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →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

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_{_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

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 15

             ↳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

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_{_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

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 15

             ↳Nar

...

               →DP Problem 16

                 ↳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

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_{_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

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 15

             ↳Nar

...

               →DP Problem 17

                 ↳Forward Instantiation Transformation

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

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')

seven 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_{_f}(X''''')))) -> SEL(s(X''), cons(Y'', n_{_f}(X''''')))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_f}(n_{_f}(X'''''))))) -> SEL(s(X''), cons(Y'', n_{_f}(n_{_f}(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_f}(n_{_g}(X'''''))))) -> SEL(s(X''), cons(Y'', n_{_f}(n_{_g}(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_g}(n_{_f}(X'''''))))) -> SEL(s(X''), cons(Y'', n_{_g}(n_{_f}(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_g}(n_{_g}(X'''''))))) -> SEL(s(X''), cons(Y'', n_{_g}(n_{_g}(X'''''))))
SEL(s(s(X'')), cons(Y, cons(Y'', n_{_g}(X''''')))) -> SEL(s(X''), cons(Y'', n_{_g}(X''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 15

             ↳Nar

...

               →DP Problem 18

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

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_{_g}(X'''))) -> SEL(X, g(X'''))

no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 15

             ↳Nar

...

               →DP Problem 19

                 ↳Argument Filtering and Ordering

Dependency Pairs:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

The following usable rules for innermost can be oriented:

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)
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

SEL > activate > f > n_{_f}
SEL > activate > f > cons
SEL > activate > f > n_{_g}
SEL > activate > g > n_{_g}
SEL > activate > g > s

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)
f(x₁) -> f(x₁)
n_{_f}(x₁) -> n_{_f}(x₁)
g(x₁) -> g(x₁)
n_{_g}(x₁) -> n_{_g}(x₁)
activate(x₁) -> activate(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

       →DP Problem 2

         ↳FwdInst

       →DP Problem 3

         ↳Nar

           →DP Problem 15

             ↳Nar

...

               →DP Problem 20

                 ↳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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:11 minutes