Term Rewriting System R:

   [X]
   f(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(X), c) -> f(X, c)
   f(c, c) -> f(a, a)

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   F(s(X), c) -> F(X, c)
   F(a, a) -> F(a, b)
   F(c, c) -> F(a, a)
   F(a, b) -> F(s(a), c)

Furthermore, R contains one SCC.


SCC1:

F(a, b) -> F(s(a), c)
F(a, a) -> F(a, b)
F(c, c) -> F(a, a)
F(s(X), c) -> F(X, c)

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(b) = 0
POL(s(x_1)) = x_1
POL(a) = 0
POL(F(x_1, x_2)) = x_1
POL(c) = 1

 resulting in one subcycle.


SCC1.Polo1:

F(s(X), c) -> F(X, c)

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)) = 1 + x_1
POL(F(x_1, x_2)) = 1 + x_1 + x_2
POL(c) = 1

The following Dependency Pairs can be deleted:

   F(s(X), c) -> F(X, c)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.551 seconds.