Termination Proof using AProVE

Term Rewriting System R:
[X]
a_{_g}(X) -> a_{_h}(X)
a_{_g}(X) -> g(X)
a_{_c} -> d
a_{_c} -> c
a_{_h}(d) -> a_{_g}(c)
a_{_h}(X) -> h(X)
mark(g(X)) -> a_{_g}(X)
mark(h(X)) -> a_{_h}(X)
mark(c) -> a_{_c}
mark(d) -> d

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A_{_G}(X) -> A_{_H}(X)
A_{_H}(d) -> A_{_G}(c)
MARK(g(X)) -> A_{_G}(X)
MARK(h(X)) -> A_{_H}(X)
MARK(c) -> A_{_C}

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Argument Filtering and Ordering

Dependency Pairs:

A_{_H}(d) -> A_{_G}(c)
A_{_G}(X) -> A_{_H}(X)

Rules:

a_{_g}(X) -> a_{_h}(X)
a_{_g}(X) -> g(X)
a_{_c} -> d
a_{_c} -> c
a_{_h}(d) -> a_{_g}(c)
a_{_h}(X) -> h(X)
mark(g(X)) -> a_{_g}(X)
mark(h(X)) -> a_{_h}(X)
mark(c) -> a_{_c}
mark(d) -> d

Strategy:

innermost

The following dependency pairs can be strictly oriented:

A_{_H}(d) -> A_{_G}(c)
A_{_G}(X) -> A_{_H}(X)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

d > A_{_G} > A_{_H}
d > c

resulting in one new DP problem.
Used Argument Filtering System:

A_{_G}(x₁) -> A_{_G}(x₁)
A_{_H}(x₁) -> A_{_H}(x₁)

     ↳DPs

       →DP Problem 1

         ↳AFS

           →DP Problem 2

             ↳Dependency Graph

Dependency Pair:

Rules:

a_{_g}(X) -> a_{_h}(X)
a_{_g}(X) -> g(X)
a_{_c} -> d
a_{_c} -> c
a_{_h}(d) -> a_{_g}(c)
a_{_h}(X) -> h(X)
mark(g(X)) -> a_{_g}(X)
mark(h(X)) -> a_{_h}(X)
mark(c) -> a_{_c}
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