Term Rewriting System R:
[x, y]
f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(x, y) -> G(x, y)
G(h(x), y) -> F(x, y)
G(h(x), y) -> G(x, y)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pairs:

G(h(x), y) -> G(x, y)
G(h(x), y) -> F(x, y)
F(x, y) -> G(x, y)

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

The following dependency pairs can be strictly oriented:

G(h(x), y) -> G(x, y)
G(h(x), y) -> F(x, y)

There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{F, G}

resulting in one new DP problem.
Used Argument Filtering System:
G(x1, x2) -> G(x1, x2)
F(x1, x2) -> F(x1, x2)
h(x1) -> h(x1)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Dependency Graph

Dependency Pair:

F(x, y) -> G(x, y)

Rules:

f(x, y) -> g(x, y)
g(h(x), y) -> h(f(x, y))
g(h(x), y) -> h(g(x, y))

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes