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
         ↳Narrowing Transformation


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




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 one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pair:

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

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