Term Rewriting System R:
[ls, a, x, k]
rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

REV(ls) -> R1(ls, empty)
R1(cons(x, k), a) -> R1(k, cons(x, a))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pair:

R1(cons(x, k), a) -> R1(k, cons(x, a))


Rules:


rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))





The following dependency pair can be strictly oriented:

R1(cons(x, k), a) -> R1(k, cons(x, a))


The following rules can be oriented:

rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(rev(x1))=  x1  
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(r1(x1, x2))=  x1 + x2  
  POL(empty)=  0  

resulting in one new DP problem.
Used Argument Filtering System:
R1(x1, x2) -> x1
cons(x1, x2) -> cons(x1, x2)
rev(x1) -> rev(x1)
r1(x1, x2) -> r1(x1, x2)


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


Dependency Pair:


Rules:


rev(ls) -> r1(ls, empty)
r1(empty, a) -> a
r1(cons(x, k), a) -> r1(k, cons(x, a))





Using the Dependency Graph resulted in no new DP problems.

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