Term Rewriting System R:

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

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   f(s(X), X) -> f(X, a(X))

where the Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(a(x_1)) = x_1
POL(f(x_1, x_2)) = 1 + x_1 + x_2
POL(c(x_1)) = 1 + x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

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

where the Polynomial interpretation:
POL(s(x_1)) = x_1
POL(f(x_1, x_2)) = 1 + x_1 + x_2
POL(c(x_1)) = 1 + x_1
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   f(X, X) -> c(X)

where the Polynomial interpretation:
POL(f(x_1, x_2)) = 1 + x_1 + x_2
POL(c(x_1)) = x_1
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.385 seconds.