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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


The following rules can be oriented:

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


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.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
s(x1) -> s(x1)
+(x1, x2) -> +(x1, x2)


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


Dependency Pair:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


The following rules can be oriented:

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


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

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


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


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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