Term Rewriting System R:
[X]
af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

MARK(f(X)) -> AF(mark(X))
MARK(f(X)) -> MARK(X)
MARK(h(X)) -> MARK(X)

Furthermore, R contains one SCC.

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

Dependency Pairs:

MARK(h(X)) -> MARK(X)
MARK(f(X)) -> MARK(X)

Rules:

af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

The following dependency pair can be strictly oriented:

MARK(h(X)) -> MARK(X)

Additionally, the following rules can be oriented:

af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(g(x1)) =  0 POL(h(x1)) =  1 + x1 POL(mark(x1)) =  x1 POL(f(x1)) =  x1 POL(a__f(x1)) =  x1

resulting in one new DP problem.

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

Dependency Pair:

MARK(f(X)) -> MARK(X)

Rules:

af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

The following dependency pair can be strictly oriented:

MARK(f(X)) -> MARK(X)

Additionally, the following rules can be oriented:

af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(g(x1)) =  0 POL(h(x1)) =  0 POL(mark(x1)) =  x1 POL(f(x1)) =  1 + x1 POL(a__f(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

af(X) -> g(h(f(X)))
af(X) -> f(X)
mark(f(X)) -> af(mark(X))
mark(g(X)) -> g(X)
mark(h(X)) -> h(mark(X))

Using the Dependency Graph resulted in no new DP problems.

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