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))


The following rules can be oriented:

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
bin > +
bin > s

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


   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