Term Rewriting System R:
[X, XS, N]
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X

Innermost Termination of R to be shown.



   R
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(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(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(nfrom(X)) -> ACTIVATE(X)
two new Dependency Pairs are created:

ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(ns(X)) -> ACTIVATE(X)


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(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(ns(X)) -> ACTIVATE(X)
three new Dependency Pairs are created:

ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 3
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(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(nfrom(nfrom(X''))) -> ACTIVATE(nfrom(X''))
two new Dependency Pairs are created:

ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 4
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(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(nfrom(ns(X''))) -> ACTIVATE(ns(X''))
three new Dependency Pairs are created:

ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 5
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(X''))) -> ACTIVATE(ns(X''))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(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(ns(ns(X''))) -> ACTIVATE(ns(X''))
three new Dependency Pairs are created:

ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 6
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(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(ns(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
two new Dependency Pairs are created:

ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 7
Forward Instantiation Transformation


Dependency Pairs:

ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(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(ns(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
three new Dependency Pairs are created:

ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 8
Argument Filtering and Ordering


Dependency Pairs:

ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X


Strategy:

innermost




The following dependency pairs can be strictly oriented:

ACTIVATE(ns(nfrom(ns(nfrom(ns(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(ns(X'''''''')))))
ACTIVATE(ns(nfrom(ns(nfrom(nfrom(X'''''''')))))) -> ACTIVATE(nfrom(ns(nfrom(nfrom(X'''''''')))))
ACTIVATE(ns(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(X''''''))))
ACTIVATE(ns(nfrom(nfrom(ns(X''''''))))) -> ACTIVATE(nfrom(nfrom(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(nfrom(nfrom(ns(X'''')))) -> ACTIVATE(nfrom(ns(X'''')))
ACTIVATE(nfrom(nfrom(nfrom(X'''')))) -> ACTIVATE(nfrom(nfrom(X'''')))
ACTIVATE(ns(nfrom(nfrom(nfrom(X''''''))))) -> ACTIVATE(nfrom(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(nfrom(nfrom(X''''''))))) -> ACTIVATE(ns(nfrom(nfrom(X''''''))))
ACTIVATE(ns(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(nfrom(ns(ns(X'''')))) -> ACTIVATE(ns(ns(X'''')))
ACTIVATE(ns(nfrom(ns(ns(X''''''))))) -> ACTIVATE(nfrom(ns(ns(X''''''))))
ACTIVATE(nfrom(ns(nfrom(ns(X''''''))))) -> ACTIVATE(ns(nfrom(ns(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)
nfrom(x1) -> nfrom(x1)
ns(x1) -> ns(x1)


   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 9
Dependency Graph


Dependency Pair:


Rules:


from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
after(0, XS) -> XS
after(s(N), cons(X, XS)) -> after(N, activate(XS))
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(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