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

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 two SCCs.


   R
DPs
       →DP Problem 1
Polynomial Ordering
       →DP Problem 2
Nar


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





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(n__from(x1))=  x1  
  POL(n__s(x1))=  1 + x1  
  POL(ACTIVATE(x1))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 3
Polynomial Ordering
       →DP Problem 2
Nar


Dependency Pair:

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





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(n__from(x1))=  1 + x1  
  POL(ACTIVATE(x1))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 3
Polo
             ...
               →DP Problem 4
Dependency Graph
       →DP Problem 2
Nar


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





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Narrowing Transformation


Dependency Pair:

AFTER(s(N), cons(X, XS)) -> AFTER(N, activate(XS))


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





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

AFTER(s(N), cons(X, XS)) -> AFTER(N, activate(XS))
three new Dependency Pairs are created:

AFTER(s(N), cons(X, nfrom(X''))) -> AFTER(N, from(activate(X'')))
AFTER(s(N), cons(X, ns(X''))) -> AFTER(N, s(activate(X'')))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 5
Narrowing Transformation


Dependency Pairs:

AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')
AFTER(s(N), cons(X, ns(X''))) -> AFTER(N, s(activate(X'')))
AFTER(s(N), cons(X, nfrom(X''))) -> AFTER(N, from(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





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

AFTER(s(N), cons(X, nfrom(X''))) -> AFTER(N, from(activate(X'')))
five new Dependency Pairs are created:

AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, cons(activate(X'''), nfrom(ns(activate(X''')))))
AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, nfrom(activate(X''')))
AFTER(s(N), cons(X, nfrom(nfrom(X''')))) -> AFTER(N, from(from(activate(X'''))))
AFTER(s(N), cons(X, nfrom(ns(X''')))) -> AFTER(N, from(s(activate(X'''))))
AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, from(X'''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 5
Nar
             ...
               →DP Problem 6
Narrowing Transformation


Dependency Pairs:

AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, from(X'''))
AFTER(s(N), cons(X, nfrom(nfrom(X''')))) -> AFTER(N, from(from(activate(X'''))))
AFTER(s(N), cons(X, nfrom(ns(X''')))) -> AFTER(N, from(s(activate(X'''))))
AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, cons(activate(X'''), nfrom(ns(activate(X''')))))
AFTER(s(N), cons(X, ns(X''))) -> AFTER(N, s(activate(X'')))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')


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





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

AFTER(s(N), cons(X, ns(X''))) -> AFTER(N, s(activate(X'')))
four new Dependency Pairs are created:

AFTER(s(N), cons(X, ns(X'''))) -> AFTER(N, ns(activate(X''')))
AFTER(s(N), cons(X, ns(nfrom(X''')))) -> AFTER(N, s(from(activate(X'''))))
AFTER(s(N), cons(X, ns(ns(X''')))) -> AFTER(N, s(s(activate(X'''))))
AFTER(s(N), cons(X, ns(X'''))) -> AFTER(N, s(X'''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 5
Nar
             ...
               →DP Problem 7
Polynomial Ordering


Dependency Pairs:

AFTER(s(N), cons(X, ns(X'''))) -> AFTER(N, s(X'''))
AFTER(s(N), cons(X, ns(ns(X''')))) -> AFTER(N, s(s(activate(X'''))))
AFTER(s(N), cons(X, ns(nfrom(X''')))) -> AFTER(N, s(from(activate(X'''))))
AFTER(s(N), cons(X, nfrom(nfrom(X''')))) -> AFTER(N, from(from(activate(X'''))))
AFTER(s(N), cons(X, nfrom(ns(X''')))) -> AFTER(N, from(s(activate(X'''))))
AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, cons(activate(X'''), nfrom(ns(activate(X''')))))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')
AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, from(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





The following dependency pairs can be strictly oriented:

AFTER(s(N), cons(X, ns(X'''))) -> AFTER(N, s(X'''))
AFTER(s(N), cons(X, ns(ns(X''')))) -> AFTER(N, s(s(activate(X'''))))
AFTER(s(N), cons(X, ns(nfrom(X''')))) -> AFTER(N, s(from(activate(X'''))))
AFTER(s(N), cons(X, nfrom(nfrom(X''')))) -> AFTER(N, from(from(activate(X'''))))
AFTER(s(N), cons(X, nfrom(ns(X''')))) -> AFTER(N, from(s(activate(X'''))))
AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, cons(activate(X'''), nfrom(ns(activate(X''')))))
AFTER(s(N), cons(X, XS')) -> AFTER(N, XS')
AFTER(s(N), cons(X, nfrom(X'''))) -> AFTER(N, from(X'''))


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(n__from(x1))=  0  
  POL(from(x1))=  0  
  POL(activate(x1))=  0  
  POL(AFTER(x1, x2))=  x1  
  POL(cons(x1, x2))=  0  
  POL(n__s(x1))=  0  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 5
Nar
             ...
               →DP Problem 8
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





Using the Dependency Graph resulted in no new DP problems.

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