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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(G(x1, x2)) =  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`
`             ↳Dependency Graph`

Dependency Pairs:

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

Using the Dependency Graph the DP problem was split into 2 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 3`
`                 ↳Instantiation Transformation`

Dependency Pair:

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

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

G(h(x, y), z) -> G(x, f(y, z))
one new Dependency Pair is created:

G(h(x'', y0), f(y'', z'')) -> G(x'', f(y0, f(y'', z'')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 5`
`                 ↳Instantiation Transformation`

Dependency Pair:

G(h(x'', y0), f(y'', z'')) -> G(x'', f(y0, f(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)

On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

G(h(x'', y0), f(y'', z'')) -> G(x'', f(y0, f(y'', z'')))
one new Dependency Pair is created:

G(h(x'''', y0''), f(y''0, f(y'''', z''''))) -> G(x'''', f(y0'', f(y''0, f(y'''', z''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 6`
`                 ↳Polynomial Ordering`

Dependency Pair:

G(h(x'''', y0''), f(y''0, f(y'''', z''''))) -> G(x'''', f(y0'', f(y''0, f(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'''', y0''), f(y''0, f(y'''', z''''))) -> G(x'''', f(y0'', f(y''0, f(y'''', z''''))))

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(G(x1, x2)) =  x1 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`
`             ↳DGraph`
`             ...`
`               →DP Problem 4`
`                 ↳Polynomial Ordering`

Dependency Pair:

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

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)

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳DGraph`
`             ...`
`               →DP Problem 7`
`                 ↳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