Term Rewriting System R:
[x]
f(g(x)) -> f(a(g(g(f(x))), g(f(x))))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(x)) -> F(a(g(g(f(x))), g(f(x))))
F(g(x)) -> F(x)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering


Dependency Pair:

F(g(x)) -> F(x)


Rule:


f(g(x)) -> f(a(g(g(f(x))), g(f(x))))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(g(x)) -> F(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:
trivial

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


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


Dependency Pair:


Rule:


f(g(x)) -> f(a(g(g(f(x))), g(f(x))))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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