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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

W(r(x)) -> W(x)
B(r(x)) -> B(x)
B(w(x)) -> W(b(x))
B(w(x)) -> B(x)

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS


Dependency Pair:

W(r(x)) -> W(x)


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




The following dependency pair can be strictly oriented:

W(r(x)) -> W(x)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(W(x1))=  x1  
  POL(r(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
W(x1) -> W(x1)
r(x1) -> r(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 4
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS


Dependency Pair:


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering
       →DP Problem 3
         ↳AFS


Dependency Pair:

B(r(x)) -> B(x)


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




The following dependency pair can be strictly oriented:

B(r(x)) -> B(x)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(B(x1))=  x1  
  POL(r(x1))=  1 + x1  

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 5
             ↳Dependency Graph
       →DP Problem 3
         ↳AFS


Dependency Pair:


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Argument Filtering and Ordering


Dependency Pair:

B(w(x)) -> B(x)


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




The following dependency pair can be strictly oriented:

B(w(x)) -> B(x)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(B(x1))=  x1  
  POL(w(x1))=  1 + x1  

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
           →DP Problem 6
             ↳Dependency Graph


Dependency Pair:


Rules:


w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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