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)


There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(+'(x1, x2))=  x1 + x2  

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


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


Dependency Pair:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 4
                 ↳Argument Filtering and Ordering


Dependency Pair:

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


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

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


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