Term Rewriting System R:

   [x, y, z]
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   +(x, +(y, z)) -> +(+(x, y), z)
   f(g(f(x))) -> f(h(s(0), x))
   f(g(h(x, y))) -> f(h(s(x), y))
   f(h(x, h(y, z))) -> f(h(+(x, y), z))

Termination of R to be shown.


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

   +(x, 0) -> x
   +(0, y) -> y

where the Polynomial interpretation:
POL(g(x_1)) = x_1
POL(s(x_1)) = x_1
POL(h(x_1, x_2)) = x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(f(x_1)) = 1 + x_1
POL(0) = 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: 

   +(x, s(y)) -> s(+(x, y))

where the Polynomial interpretation:
POL(g(x_1)) = 1 + 2*x_1
POL(s(x_1)) = 1 + x_1
POL(h(x_1, x_2)) = x_1 + 2*x_2
POL(+(x_1, x_2)) = x_1 + 2*x_2
POL(f(x_1)) = x_1
POL(0) = 0
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: 

   +(x, +(y, z)) -> +(+(x, y), z)
   f(g(f(x))) -> f(h(s(0), x))
   f(g(h(x, y))) -> f(h(s(x), y))
   f(h(x, h(y, z))) -> f(h(+(x, y), z))

where the Polynomial interpretation:
POL(g(x_1)) = 2*x_1
POL(s(x_1)) = x_1
POL(h(x_1, x_2)) = 1 + x_1 + 2*x_2
POL(+(x_1, x_2)) = 1 + x_1 + 2*x_2
POL(f(x_1)) = 1 + x_1
POL(0) = 0
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: 

   +(s(x), y) -> s(+(x, y))

where the Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(+(x_1, x_2)) = 2*x_1 + x_2
was used. 



All Rules of R can be deleted.



Termination of R successfully shown.


Duration: 0.497 seconds.