Term Rewriting System R:
[x, y]
average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

AVERAGE(s(x), y) -> AVERAGE(x, s(y))
AVERAGE(x, s(s(s(y)))) -> AVERAGE(s(x), y)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

AVERAGE(x, s(s(s(y)))) -> AVERAGE(s(x), y)
AVERAGE(s(x), y) -> AVERAGE(x, s(y))


Rules:


average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)





The following dependency pair can be strictly oriented:

AVERAGE(x, s(s(s(y)))) -> AVERAGE(s(x), y)


The following rules can be oriented:

average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(average(x1, x2))=  x1 + x2  
  POL(AVERAGE(x1, x2))=  1 + x1 + x2  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
AVERAGE(x1, x2) -> AVERAGE(x1, x2)
s(x1) -> s(x1)
average(x1, x2) -> average(x1, x2)


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


Dependency Pair:

AVERAGE(s(x), y) -> AVERAGE(x, s(y))


Rules:


average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)





The following dependency pair can be strictly oriented:

AVERAGE(s(x), y) -> AVERAGE(x, s(y))


The following rules can be oriented:

average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(average(x1, x2))=  x1 + x2  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
AVERAGE(x1, x2) -> x1
s(x1) -> s(x1)
average(x1, x2) -> average(x1, x2)


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


Dependency Pair:


Rules:


average(s(x), y) -> average(x, s(y))
average(x, s(s(s(y)))) -> s(average(s(x), y))
average(0, 0) -> 0
average(0, s(0)) -> 0
average(0, s(s(0))) -> s(0)





Using the Dependency Graph resulted in no new DP problems.

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