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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

G(f(x, y), z) -> G(y, z)
G(h(x, y), z) -> G(x, f(y, z))
G(x, h(y, z)) -> G(x, y)

Furthermore, R contains one SCC.

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

Dependency Pairs:

G(x, h(y, z)) -> G(x, y)
G(h(x, y), z) -> G(x, f(y, z))
G(f(x, y), z) -> G(y, z)

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

G(x, h(y, z)) -> G(x, y)
G(h(x, y), z) -> G(x, f(y, z))

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(G(x1, x2)) =  x1 + x2 POL(h(x1, x2)) =  1 + x1 + x2 POL(f(x1, x2)) =  x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Argument Filtering and Ordering`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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(G(x1, x2)) =  x1 + x2 POL(f(x1, x2)) =  1 + x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳AFS`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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