Term Rewriting System R:
[X, XS, X1, X2]
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ATAIL(cons(X, XS)) -> MARK(XS)
MARK(zeros) -> AZEROS
MARK(tail(X)) -> ATAIL(mark(X))
MARK(tail(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(tail(X)) -> MARK(X)
MARK(tail(X)) -> ATAIL(mark(X))
ATAIL(cons(X, XS)) -> MARK(XS)


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0





The following dependency pairs can be strictly oriented:

MARK(tail(X)) -> MARK(X)
MARK(tail(X)) -> ATAIL(mark(X))


Additionally, the following usable rules using the Ce-refinement can be oriented:

mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
azeros -> cons(0, zeros)
azeros -> zeros


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(MARK(x1))=  x1  
  POL(a__zeros)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(tail(x1))=  1 + x1  
  POL(a__tail(x1))=  1 + x1  
  POL(mark(x1))=  x1  
  POL(zeros)=  0  
  POL(A__TAIL(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Dependency Graph


Dependency Pairs:

MARK(cons(X1, X2)) -> MARK(X1)
ATAIL(cons(X, XS)) -> MARK(XS)


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0





Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 3
                 ↳Polynomial Ordering


Dependency Pair:

MARK(cons(X1, X2)) -> MARK(X1)


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0





The following dependency pair can be strictly oriented:

MARK(cons(X1, X2)) -> MARK(X1)


There are no usable rules using the Ce-refinement that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 4
                 ↳Dependency Graph


Dependency Pair:


Rules:


azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0





Using the Dependency Graph resulted in no new DP problems.

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