Term Rewriting System R:

   [x, y]
   f(x, c(y)) -> f(x, s(f(y, y)))
   f(s(x), y) -> f(x, s(c(y)))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   F(x, c(y)) -> F(x, s(f(y, y)))
   F(x, c(y)) -> F(y, y)
   F(s(x), y) -> F(x, s(c(y)))

Furthermore, R contains two SCCs.


SCC1:

F(s(x), y) -> F(x, s(c(y)))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = x_1
POL(F(x_1, x_2)) = 1 + x_1 + x_2
POL(c(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   f(x, c(y)) -> f(x, s(f(y, y)))
   f(s(x), y) -> f(x, s(c(y)))


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

F(s(x), y) -> F(x, s(c(y)))

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC1.MRR1.NOC1:

F(s(x), y) -> F(x, s(c(y)))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(F(x_1, x_2)) = x_1
POL(c(x_1)) = 0

 resulting in no subcycles.


SCC2:

F(x, c(y)) -> F(y, y)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(F(x_1, x_2)) = 1 + x_2
POL(c(x_1)) = 1 + x_1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.514 seconds.