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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

BIN(s(x), s(y)) -> BIN(x, s(y))
BIN(s(x), s(y)) -> BIN(x, y)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

BIN(s(x), s(y)) -> BIN(x, y)
BIN(s(x), s(y)) -> BIN(x, s(y))


Rules:


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





The following dependency pairs can be strictly oriented:

BIN(s(x), s(y)) -> BIN(x, y)
BIN(s(x), s(y)) -> BIN(x, s(y))


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(BIN(x1, x2))=  x1 + x2  

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


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


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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