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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

+'(s(x), s(y)) -> +'(s(x), +(y, 0))
+'(s(x), s(y)) -> +'(y, 0)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pair:

+'(s(x), s(y)) -> +'(s(x), +(y, 0))


Rules:


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





The following dependency pair can be strictly oriented:

+'(s(x), s(y)) -> +'(s(x), +(y, 0))


Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

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


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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