Term Rewriting System R:
[X]
ag(X) -> ah(X)
ag(X) -> g(X)
ac -> d
ac -> c
ah(d) -> ag(c)
ah(X) -> h(X)
mark(g(X)) -> ag(X)
mark(h(X)) -> ah(X)
mark(c) -> ac
mark(d) -> d

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AG(X) -> AH(X)
AH(d) -> AG(c)
MARK(g(X)) -> AG(X)
MARK(h(X)) -> AH(X)
MARK(c) -> AC

Furthermore, R contains one SCC.

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

Dependency Pairs:

AH(d) -> AG(c)
AG(X) -> AH(X)

Rules:

ag(X) -> ah(X)
ag(X) -> g(X)
ac -> d
ac -> c
ah(d) -> ag(c)
ah(X) -> h(X)
mark(g(X)) -> ag(X)
mark(h(X)) -> ah(X)
mark(c) -> ac
mark(d) -> d

The following dependency pair can be strictly oriented:

AG(X) -> AH(X)

The following rules can be oriented:

ag(X) -> ah(X)
ag(X) -> g(X)
ac -> d
ac -> c
ah(d) -> ag(c)
ah(X) -> h(X)
mark(g(X)) -> ag(X)
mark(h(X)) -> ah(X)
mark(c) -> ac
mark(d) -> d

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(A__G(x1)) =  1 + x1 POL(a__g(x1)) =  x1 POL(c) =  0 POL(a__c) =  1 POL(g(x1)) =  x1 POL(d) =  1 POL(h(x1)) =  x1 POL(mark(x1)) =  1 + x1 POL(A__H(x1)) =  x1 POL(a__h(x1)) =  x1

resulting in one new DP problem.
Used Argument Filtering System:
AG(x1) -> AG(x1)
AH(x1) -> AH(x1)
ag(x1) -> ag(x1)
ah(x1) -> ah(x1)
g(x1) -> g(x1)
ac -> ac
h(x1) -> h(x1)
mark(x1) -> mark(x1)

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

Dependency Pair:

AH(d) -> AG(c)

Rules:

ag(X) -> ah(X)
ag(X) -> g(X)
ac -> d
ac -> c
ah(d) -> ag(c)
ah(X) -> h(X)
mark(g(X)) -> ag(X)
mark(h(X)) -> ah(X)
mark(c) -> ac
mark(d) -> d

Using the Dependency Graph resulted in no new DP problems.

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