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
         ↳Argument Filtering and 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 pairs can be strictly oriented:

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


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: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
mark > af > h
mark > af > f
mark > af > g

resulting in one new DP problem.
Used Argument Filtering System:
MARK(x1) -> MARK(x1)
f(x1) -> f(x1)
h(x1) -> h(x1)
af(x1) -> af(x1)
g(x1) -> g(x1)
mark(x1) -> mark(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳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