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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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

F(f(a, a), a) -> F(f(a, f(a, f(a, a))), a)
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.


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