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

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pairs:

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

Rule:

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

The following dependency pair can be strictly oriented:

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

The following rule can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
a > f

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
f(x1, x2) -> f

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

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

Rule:

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

Termination of R could not be shown.
Duration:
0:00 minutes