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
         ↳Narrowing Transformation


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)))





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

+'(s(x), s(y)) -> +'(s(x), +(y, 0))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

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


Rules:


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




Termination of R could not be shown.
Duration:
0:00 minutes