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

Innermost 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 one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

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

Strategy:

innermost

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

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

two new Dependency Pairs are created:

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

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

Strategy:

innermost

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

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

two new Dependency Pairs are created:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

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

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

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

two new Dependency Pairs are created:

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

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

three new Dependency Pairs are created:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS 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_{_from}(x₁) -> n_{_from}(x₁)
n_{_s}(x₁) -> n_{_s}(x₁)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 9

                 ↳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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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