Term Rewriting System R:

   [X, Z, Y]
   h(X, Z) -> f(X, s(X), Z)
   f(X, Y, g(X, Y)) -> h(0, g(X, Y))
   g(0, Y) -> 0
   g(X, s(Y)) -> g(X, Y)

Termination of R to be shown.


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

   g(0, Y) -> 0

where the Polynomial interpretation:
POL(g(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = x_1
POL(h(x_1, x_2)) = 2*x_1 + x_2
POL(f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(0) = 0
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(X, Y, g(X, Y)) -> H(0, g(X, Y))
   G(X, s(Y)) -> G(X, Y)
   H(X, Z) -> F(X, s(X), Z)

Furthermore, R contains two SCCs.


SCC1:

G(X, s(Y)) -> G(X, Y)

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(G(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   G(X, s(Y)) -> G(X, Y)



 This transformation is resulting in no new subcycles.


SCC2:

H(X, Z) -> F(X, s(X), Z)
F(X, Y, g(X, Y)) -> H(0, g(X, Y))

Found an infinite P-chain over R:
P = H(X, Z) -> F(X, s(X), Z)
F(X, Y, g(X, Y)) -> H(0, g(X, Y))
R = [f(X, Y, g(X, Y)) -> h(0, g(X, Y)), g(X, s(Y)) -> g(X, Y), h(X, Z) -> f(X, s(X), Z)]
s = H(0, g(0, s(0))) evaluates to t = H(0, g(0, s(0)))
Thus, s starts an infinite reduction.


Non-Termination of R could be shown.

Duration: 0.593 seconds.