Termination Proof using AProVE

Term Rewriting System R:
[X]
f(0) -> cons(0, n_{_f}(n_{_s}(n_₀)))
f(s(0)) -> f(p(s(0)))
f(X) -> n_{_f}(X)
p(s(0)) -> 0
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

F(s(0)) -> F(p(s(0)))
F(s(0)) -> P(s(0))
ACTIVATE(n_{_f}(X)) -> F(activate(X))
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_s}(X)) -> S(activate(X))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_₀) -> 0'

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rules:

f(0) -> cons(0, n_{_f}(n_{_s}(n_₀)))
f(s(0)) -> f(p(s(0)))
f(X) -> n_{_f}(X)
p(s(0)) -> 0
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
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_{_s}(X''))) -> ACTIVATE(n_{_s}(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_{_f}(n_{_s}(X''))) -> ACTIVATE(n_{_s}(X''))
ACTIVATE(n_{_f}(n_{_f}(X''))) -> ACTIVATE(n_{_f}(X''))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Rules:

f(0) -> cons(0, n_{_f}(n_{_s}(n_₀)))
f(s(0)) -> f(p(s(0)))
f(X) -> n_{_f}(X)
p(s(0)) -> 0
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
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_{_s}(X)) -> ACTIVATE(X)

three new Dependency Pairs are created:

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

Rules:

f(0) -> cons(0, n_{_f}(n_{_s}(n_₀)))
f(s(0)) -> f(p(s(0)))
f(X) -> n_{_f}(X)
p(s(0)) -> 0
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
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_{_s}(X'''')))) -> ACTIVATE(n_{_f}(n_{_s}(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_{_f}(n_{_f}(n_{_s}(X'''')))) -> ACTIVATE(n_{_f}(n_{_s}(X'''')))
ACTIVATE(n_{_f}(n_{_f}(n_{_f}(X'''')))) -> ACTIVATE(n_{_f}(n_{_f}(X'''')))
ACTIVATE(n_{_s}(n_{_f}(n_{_f}(X'''')))) -> ACTIVATE(n_{_f}(n_{_f}(X'''')))
ACTIVATE(n_{_s}(n_{_s}(X''))) -> ACTIVATE(n_{_s}(X''))
ACTIVATE(n_{_f}(n_{_s}(X''))) -> ACTIVATE(n_{_s}(X''))
ACTIVATE(n_{_s}(n_{_f}(n_{_s}(X'''')))) -> ACTIVATE(n_{_f}(n_{_s}(X'''')))

Rules:

f(0) -> cons(0, n_{_f}(n_{_s}(n_₀)))
f(s(0)) -> f(p(s(0)))
f(X) -> n_{_f}(X)
p(s(0)) -> 0
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
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_{_s}(X''))) -> ACTIVATE(n_{_s}(X''))

three new Dependency Pairs are created:

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

Rules:

f(0) -> cons(0, n_{_f}(n_{_s}(n_₀)))
f(s(0)) -> f(p(s(0)))
f(X) -> n_{_f}(X)
p(s(0)) -> 0
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
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_{_s}(n_{_s}(X''))) -> ACTIVATE(n_{_s}(X''))

three new Dependency Pairs are created:

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

Rules:

f(0) -> cons(0, n_{_f}(n_{_s}(n_₀)))
f(s(0)) -> f(p(s(0)))
f(X) -> n_{_f}(X)
p(s(0)) -> 0
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
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_{_s}(n_{_f}(n_{_f}(X'''')))) -> ACTIVATE(n_{_f}(n_{_f}(X'''')))

two new Dependency Pairs are created:

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

Rules:

f(0) -> cons(0, n_{_f}(n_{_s}(n_₀)))
f(s(0)) -> f(p(s(0)))
f(X) -> n_{_f}(X)
p(s(0)) -> 0
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
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_{_s}(n_{_f}(n_{_s}(X'''')))) -> ACTIVATE(n_{_f}(n_{_s}(X'''')))

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 8

                 ↳Argument Filtering and Ordering

Dependency Pairs:

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

Rules:

f(0) -> cons(0, n_{_f}(n_{_s}(n_₀)))
f(s(0)) -> f(p(s(0)))
f(X) -> n_{_f}(X)
p(s(0)) -> 0
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

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

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 9

                 ↳Dependency Graph

Dependency Pair:

Rules:

f(0) -> cons(0, n_{_f}(n_{_s}(n_₀)))
f(s(0)) -> f(p(s(0)))
f(X) -> n_{_f}(X)
p(s(0)) -> 0
s(X) -> n_{_s}(X)
0 -> n_₀
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_₀) -> 0
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