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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

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


Rules:


+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)





The following dependency pairs can be strictly oriented:

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


Additionally, the following rules can be oriented:

+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)


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

resulting in one new DP problem. 



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


Dependency Pairs:

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


Rules:


+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)





The following dependency pair can be strictly oriented:

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


Additionally, the following rules can be oriented:

+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Polynomial Ordering


Dependency Pairs:

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


Rules:


+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)





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:

+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)


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


+(x, +(y, z)) -> +(+(x, y), z)
+(*(x, y), +(x, z)) -> *(x, +(y, z))
+(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u)





Using the Dependency Graph resulted in no new DP problems.

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