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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Argument Filtering and Ordering


Dependency Pair:

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


Rule:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rule for innermost w.r.t. to the AFS can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(b(x1))=  x1  
  POL(a(x1))=  1 + x1  
  POL(A(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
A(x1) -> A(x1)
b(x1) -> b(x1)
a(x1) -> a(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳AFS
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rule:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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