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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(f(x1)) =  1 + x1

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

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

Dependency Pair:

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

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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