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

Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pairs:

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

Rule:

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

The following dependency pair can be strictly oriented:

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

The following rule can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(a) =  1 POL(f) =  0

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

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

The following remains to be proven:
Dependency Pairs:

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

Rule:

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

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