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

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


Rule:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rule for innermost can be oriented:

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


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
             ↳Narrowing Transformation


Dependency Pair:

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


Rule:


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


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(a, f(a, x)) -> F(a, f(f(a, a), f(a, x)))
one new Dependency Pair is created:

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

The transformation is resulting in no new DP problems.


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