Term Rewriting System R:

   [X]
   f(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(X)

Termination of R to be shown.


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

   f(c(X)) -> X
   f(d(X)) -> X

where the Polynomial interpretation:
POL(b) = 0
POL(g(x_1)) = x_1
POL(e(x_1)) = x_1
POL(d(x_1)) = x_1
POL(a) = 0
POL(f(x_1)) = 1 + x_1
POL(c(x_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: 

   e(g(X)) -> e(X)

where the Polynomial interpretation:
POL(b) = 0
POL(g(x_1)) = 1 + x_1
POL(e(x_1)) = x_1
POL(d(x_1)) = x_1
POL(a) = 0
POL(f(x_1)) = x_1
POL(c(x_1)) = x_1
was used. 



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


R contains the following Dependency Pairs: 


   F(c(b)) -> F(d(a))
   F(a) -> F(d(a))
   F(c(a)) -> F(d(b))
   F(a) -> F(c(a))

R contains no SCCs.




Termination of R successfully shown.


Duration: 0.477 seconds.