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)

Innermost Termination of R to be shown.


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 one SCC.


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.




Innermost Termination of R successfully shown.


Duration: 0.420 seconds.