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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs. 


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


Dependency Pair:

S(+(0, y)) -> S(y)


Rules:


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





The following dependency pair can be strictly oriented:

S(+(0, y)) -> S(y)


The following rules can be oriented:

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


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

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


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


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


The following rules can be oriented:

+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, s(y)) -> s(y)
s(+(0, y)) -> 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
         ↳AFS
           →DP Problem 4
             ↳Dependency Graph


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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