Term Rewriting System R:
[x, y, z]
f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
g(0, 1) -> 0
g(0, 1) -> 1
h(g(x, y)) -> h(x)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(0, 1, g(x, y), z) -> F(g(x, y), g(x, y), g(x, y), h(x))
F(0, 1, g(x, y), z) -> H(x)
H(g(x, y)) -> H(x)

Furthermore, R contains one SCC.

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

Dependency Pair:

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

Rules:

f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
g(0, 1) -> 0
g(0, 1) -> 1
h(g(x, y)) -> h(x)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1, x2)) =  1 + x1 + x2 POL(H(x1)) =  x1

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

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

Dependency Pair:

Rules:

f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
g(0, 1) -> 0
g(0, 1) -> 1
h(g(x, y)) -> h(x)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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