Term Rewriting System R:

   [x]
   d(x) -> e(u(x))
   d(u(x)) -> c(x)
   c(u(x)) -> b(x)
   v(e(x)) -> x
   b(u(x)) -> a(e(x))

Termination of R to be shown.


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

   v(e(x)) -> x

where the Polynomial interpretation:
POL(b(x_1)) = x_1
POL(v(x_1)) = 1 + x_1
POL(e(x_1)) = x_1
POL(d(x_1)) = x_1
POL(a(x_1)) = x_1
POL(u(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.


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

   d(x) -> e(u(x))
   d(u(x)) -> c(x)

where the Polynomial interpretation:
POL(b(x_1)) = x_1
POL(e(x_1)) = x_1
POL(d(x_1)) = 1 + x_1
POL(a(x_1)) = x_1
POL(u(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.


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

   c(u(x)) -> b(x)
   b(u(x)) -> a(e(x))

where the Polynomial interpretation:
POL(b(x_1)) = x_1
POL(e(x_1)) = x_1
POL(a(x_1)) = x_1
POL(c(x_1)) = x_1
POL(u(x_1)) = 1 + x_1
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.445 seconds.