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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(+(x, 0)) -> F(x)
+'(x, +(y, z)) -> +'(+(x, y), z)
+'(x, +(y, z)) -> +'(x, y)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo


Dependency Pair:

F(+(x, 0)) -> F(x)


Rules:


f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)





The following dependency pair can be strictly oriented:

F(+(x, 0)) -> F(x)


Additionally, the following rules can be oriented:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)


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

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering


Dependency Pairs:

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


Rules:


f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)





The following dependency pairs can be strictly oriented:

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


Additionally, the following rules can be oriented:

f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 4
             ↳Dependency Graph


Dependency Pair:


Rules:


f(+(x, 0)) -> f(x)
+(x, +(y, z)) -> +(+(x, y), z)





Using the Dependency Graph resulted in no new DP problems.

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