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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

A(x, g) -> A(g, a(f, x))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in no new DP problems.


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