Term Rewriting System R:

   [X, Y]
   f(g(X), Y) -> f(X, f(g(X), Y))

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   F(g(X), Y) -> F(X, f(g(X), Y))
   F(g(X), Y) -> F(g(X), Y)

Furthermore, R contains one SCC.


SCC1:

F(g(X), Y) -> F(g(X), Y)
F(g(X), Y) -> F(X, f(g(X), Y))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(g(x_1)) = 1 + x_1
POL(F(x_1, x_2)) = x_1
POL(f(x_1, x_2)) = 0

 resulting in one subcycle.


SCC1.Polo1:

F(g(X), Y) -> F(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)) = 1 + x_1
POL(F(x_1, x_2)) = 1 + x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   f(g(X), Y) -> f(X, f(g(X), Y))


 This transformation is resulting in one new subcycle:


SCC1.Polo1.MRR1:

F(g(X), Y) -> F(g(X), Y)

Found an infinite P-chain over R:
P = F(g(X), Y) -> F(g(X), Y)
R = []
s = F(g(X'), Y') evaluates to t = F(g(X'), Y')
Thus, s starts an infinite reduction.


Non-Termination of R could be shown.

Duration: 0.527 seconds.