Term Rewriting System R:
[X]
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(X)) -> F(f(X))
F(g(X)) -> F(X)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

F(g(X)) -> F(X)
F(g(X)) -> F(f(X))


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(g(X)) -> F(X)
F(g(X)) -> F(f(X))


The following usable rules for innermost w.r.t. to the AFS can be oriented:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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


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


Dependency Pair:


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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