Termination Proof using AProVE

Term Rewriting System R:
[X]
f(X) -> g(n_{_h}(n_{_f}(X)))
f(X) -> n_{_f}(X)
h(X) -> n_{_h}(X)
activate(n_{_h}(X)) -> h(activate(X))
activate(n_{_f}(X)) -> f(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

ACTIVATE(n_{_h}(X)) -> H(activate(X))
ACTIVATE(n_{_h}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_f}(X)) -> F(activate(X))
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(X) -> g(n_{_h}(n_{_f}(X)))
f(X) -> n_{_f}(X)
h(X) -> n_{_h}(X)
activate(n_{_h}(X)) -> h(activate(X))
activate(n_{_f}(X)) -> f(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_{_h}(X)) -> ACTIVATE(X)

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

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

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(X) -> g(n_{_h}(n_{_f}(X)))
f(X) -> n_{_f}(X)
h(X) -> n_{_h}(X)
activate(n_{_h}(X)) -> h(activate(X))
activate(n_{_f}(X)) -> f(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_{_h}(n_{_h}(X''))) -> ACTIVATE(n_{_h}(X''))

two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(X) -> g(n_{_h}(n_{_f}(X)))
f(X) -> n_{_f}(X)
h(X) -> n_{_h}(X)
activate(n_{_h}(X)) -> h(activate(X))
activate(n_{_f}(X)) -> f(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_{_h}(n_{_f}(X''))) -> ACTIVATE(n_{_f}(X''))

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(X) -> g(n_{_h}(n_{_f}(X)))
f(X) -> n_{_f}(X)
h(X) -> n_{_h}(X)
activate(n_{_h}(X)) -> h(activate(X))
activate(n_{_f}(X)) -> f(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''))

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(X) -> g(n_{_h}(n_{_f}(X)))
f(X) -> n_{_f}(X)
h(X) -> n_{_h}(X)
activate(n_{_h}(X)) -> h(activate(X))
activate(n_{_f}(X)) -> f(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_{_h}(n_{_h}(X'''')))) -> ACTIVATE(n_{_h}(n_{_h}(X'''')))

two new Dependency Pairs are created:

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

Dependency Pairs:

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

Rules:

f(X) -> g(n_{_h}(n_{_f}(X)))
f(X) -> n_{_f}(X)
h(X) -> n_{_h}(X)
activate(n_{_h}(X)) -> h(activate(X))
activate(n_{_f}(X)) -> f(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_{_h}(n_{_f}(X'''')))) -> ACTIVATE(n_{_h}(n_{_f}(X'''')))

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 8

                 ↳Polynomial Ordering

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__h(x₁)) = x₁

POL(n__f(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 9

                 ↳Dependency Graph

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

Using the Dependency Graph the DP problem was split into 1 DP problems.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 10

                 ↳Polynomial Ordering

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(n__h(x₁)) = 1 + x₁

POL(ACTIVATE(x₁)) = 1 + x₁

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 11

                 ↳Dependency Graph

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

POL(n__h(x₁))	= x₁
POL(n__f(x₁))	= 1 + x₁
POL(ACTIVATE(x₁))	= 1 + x₁