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
         ↳Polynomial 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)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem. 



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


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)





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

AVERAGE(s(x), y) -> AVERAGE(x, s(y))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 3
                 ↳Instantiation Transformation


Dependency Pair:

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





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

AVERAGE(s(x''), s(y'')) -> AVERAGE(x'', s(s(y'')))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 4
                 ↳Polynomial Ordering


Dependency Pair:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 5
                 ↳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