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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

H(f(x), y) -> G(x, y)
G(x, y) -> H(x, y)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`

Dependency Pairs:

G(x, y) -> H(x, y)
H(f(x), y) -> G(x, y)

Rules:

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

The following dependency pair can be strictly oriented:

H(f(x), y) -> G(x, y)

The following rules can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{h, g} > f
{H, G}

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

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

Dependency Pair:

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

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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