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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{A, B}
{a, b}

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


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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{A, B}

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


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


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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