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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

A(b(x)) -> A(a(x))
A(b(x)) -> A(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

A(b(x)) -> A(x)
A(b(x)) -> A(a(x))


Rule:


a(b(x)) -> b(b(a(a(x))))


Strategy:

innermost




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

A(b(x)) -> A(a(x))
one new Dependency Pair is created:

A(b(b(x''))) -> A(b(b(a(a(x'')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pairs:

A(b(b(x''))) -> A(b(b(a(a(x'')))))
A(b(x)) -> A(x)


Rule:


a(b(x)) -> b(b(a(a(x))))


Strategy:

innermost




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

A(b(x)) -> A(x)
two new Dependency Pairs are created:

A(b(b(x''))) -> A(b(x''))
A(b(b(b(x'''')))) -> A(b(b(x'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Forward Instantiation Transformation


Dependency Pairs:

A(b(b(b(x'''')))) -> A(b(b(x'''')))
A(b(b(x''))) -> A(b(x''))
A(b(b(x''))) -> A(b(b(a(a(x'')))))


Rule:


a(b(x)) -> b(b(a(a(x))))


Strategy:

innermost




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

A(b(b(x''))) -> A(b(x''))
two new Dependency Pairs are created:

A(b(b(b(x'''')))) -> A(b(b(x'''')))
A(b(b(b(b(x''''''))))) -> A(b(b(b(x''''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

A(b(b(b(b(x''''''))))) -> A(b(b(b(x''''''))))
A(b(b(b(x'''')))) -> A(b(b(x'''')))
A(b(b(x''))) -> A(b(b(a(a(x'')))))
A(b(b(b(x'''')))) -> A(b(b(x'''')))


Rule:


a(b(x)) -> b(b(a(a(x))))


Strategy:

innermost



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