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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(h(x1)) =  1 + x1

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

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

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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