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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs. 


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


Dependency Pair:

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


Rules:


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


Strategy:

innermost




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

A(g, a(f, a(g, x))) -> A(g, a(f, a(f, a(g, x))))
two new Dependency Pairs are created:

A(g, a(f, a(g, a(f, x'')))) -> A(g, a(f, a(f, a(g, a(g, a(f, x''))))))
A(g, a(f, a(g, a(f, a(g, x''))))) -> A(g, a(f, a(f, a(g, a(f, a(f, a(g, x'')))))))

The transformation is resulting in no new DP problems. 



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


Dependency Pair:

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


Rules:


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


Strategy:

innermost




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

A(f, a(g, a(f, x))) -> A(f, a(g, a(g, a(f, x))))
two new Dependency Pairs are created:

A(f, a(g, a(f, a(g, x'')))) -> A(f, a(g, a(g, a(f, a(f, a(g, x''))))))
A(f, a(g, a(f, a(g, a(f, x''))))) -> A(f, a(g, a(g, a(f, a(g, a(g, a(f, x'')))))))

The transformation is resulting in no new DP problems.


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