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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


a(f, a(f, x)) -> a(x, x)
a(h, x) -> 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(h, x) -> A(f, a(g, a(f, x)))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

A(h, x) -> A(f, x)
one new Dependency Pair is created:

A(h, h) -> A(f, h)

The transformation is resulting in no new DP problems.


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