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

Innermost 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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
d > AG > AH
d > c

resulting in one new DP problem.
Used Argument Filtering System:
AG(x1) -> AG(x1)
AH(x1) -> AH(x1)

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

Dependency Pair:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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