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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pair:

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


Rules:


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


Strategy:

innermost




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))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

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


Strategy:

innermost




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(X''))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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

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

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

The transformation is resulting in two new DP problems: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Polynomial Ordering


Dependency Pair:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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(AFTER(x1, x2))=  x1  
  POL(cons(x1, x2))=  0  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


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

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

resulting in one new DP problem.


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