Term Rewriting System R:
[X, Y]
f(g(X), Y) -> f(X, f(g(X), Y))

Termination of R to be shown.

R
Dependency Pair Analysis

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.

R
DPs
→DP Problem 1
Polynomial Ordering

Dependency Pairs:

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

Rule:

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

The following dependency pair can be strictly oriented:

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

Additionally, the following usable rule using the Ce-refinement can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(f(x1, x2)) =  0 POL(F(x1, x2)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

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

Rule:

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

Termination of R could not be shown.
Duration:
0:00 minutes