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)

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`
`         ↳Polynomial 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)

The following dependency pair can be strictly oriented:

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

Additionally, the following rules can be oriented:

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)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1, x2)) =  x2 POL(G(x1, x2)) =  x2 POL(h(x1, x2)) =  1 + x1 POL(f(x1, x2)) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polynomial Ordering`

Dependency Pairs:

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)

The following dependency pair can be strictly oriented:

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

Additionally, the following rules can be oriented:

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)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1, x2)) =  x2 POL(G(x1, x2)) =  x1 POL(h(x1, x2)) =  1 + x1 POL(f(x1, x2)) =  x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial 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)

The following dependency pair can be strictly oriented:

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

Additionally, the following rules can be oriented:

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)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1, x2)) =  x1 POL(G(x1, x2)) =  1 + x1 POL(h(x1, x2)) =  x1 POL(f(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 4`
`                 ↳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)

Using the Dependency Graph resulted in no new DP problems.

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