Term Rewriting System R:
[X, Y, X1, X2]
af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AF(g(X), Y) -> AF(mark(X), f(g(X), Y))
AF(g(X), Y) -> MARK(X)
MARK(f(X1, X2)) -> AF(mark(X1), X2)
MARK(f(X1, X2)) -> MARK(X1)
MARK(g(X)) -> MARK(X)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pairs:

MARK(g(X)) -> MARK(X)
MARK(f(X1, X2)) -> MARK(X1)
MARK(f(X1, X2)) -> AF(mark(X1), X2)
AF(g(X), Y) -> MARK(X)
AF(g(X), Y) -> AF(mark(X), f(g(X), Y))

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

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

MARK(f(X1, X2)) -> AF(mark(X1), X2)
two new Dependency Pairs are created:

MARK(f(f(X1'', X2''), X2)) -> AF(af(mark(X1''), X2''), X2)
MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Narrowing Transformation`

Dependency Pairs:

MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)
AF(g(X), Y) -> MARK(X)
AF(g(X), Y) -> AF(mark(X), f(g(X), Y))
MARK(f(f(X1'', X2''), X2)) -> AF(af(mark(X1''), X2''), X2)
MARK(f(X1, X2)) -> MARK(X1)
MARK(g(X)) -> MARK(X)

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

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

MARK(f(f(X1'', X2''), X2)) -> AF(af(mark(X1''), X2''), X2)
three new Dependency Pairs are created:

MARK(f(f(X1''', X2'''), X2)) -> AF(f(mark(X1'''), X2'''), X2)
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pairs:

MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
MARK(g(X)) -> MARK(X)
MARK(f(X1, X2)) -> MARK(X1)
AF(g(X), Y) -> MARK(X)
AF(g(X), Y) -> AF(mark(X), f(g(X), Y))
MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

The following dependency pairs can be strictly oriented:

MARK(g(X)) -> MARK(X)
AF(g(X), Y) -> MARK(X)
AF(g(X), Y) -> AF(mark(X), f(g(X), Y))

Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))
af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MARK(x1)) =  x1 POL(g(x1)) =  1 + x1 POL(A__F(x1, x2)) =  x1 POL(mark(x1)) =  x1 POL(f(x1, x2)) =  x1 POL(a__f(x1, x2)) =  x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pairs:

MARK(f(f(g(X'), X2''), X2)) -> AF(af(g(mark(X')), X2''), X2)
MARK(f(f(f(X1', X2'''), X2''), X2)) -> AF(af(af(mark(X1'), X2'''), X2''), X2)
MARK(f(X1, X2)) -> MARK(X1)
MARK(f(g(X'), X2)) -> AF(g(mark(X')), X2)

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Using the Dependency Graph the DP problem was split into 1 DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Polynomial Ordering`

Dependency Pair:

MARK(f(X1, X2)) -> MARK(X1)

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

The following dependency pair can be strictly oriented:

MARK(f(X1, X2)) -> MARK(X1)

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Nar`
`             ...`
`               →DP Problem 6`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

af(g(X), Y) -> af(mark(X), f(g(X), Y))
af(X1, X2) -> f(X1, X2)
mark(f(X1, X2)) -> af(mark(X1), X2)
mark(g(X)) -> g(mark(X))

Using the Dependency Graph resulted in no new DP problems.

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