Term Rewriting System R:
[x]
+(p1, p1) -> p2
+(p1, +(p2, p2)) -> p5
+(p5, p5) -> p10
+(+(x, y), z) -> +(x, +(y, z))
+(p1, +(p1, x)) -> +(p2, x)
+(p1, +(p2, +(p2, x))) -> +(p5, x)
+(p2, p1) -> +(p1, p2)
+(p2, +(p1, x)) -> +(p1, +(p2, x))
+(p2, +(p2, p2)) -> +(p1, p5)
+(p2, +(p2, +(p2, x))) -> +(p1, +(p5, x))
+(p5, p1) -> +(p1, p5)
+(p5, +(p1, x)) -> +(p1, +(p5, x))
+(p5, p2) -> +(p2, p5)
+(p5, +(p2, x)) -> +(p2, +(p5, x))
+(p5, +(p5, x)) -> +(p10, x)
+(p10, p1) -> +(p1, p10)
+(p10, +(p1, x)) -> +(p1, +(p10, x))
+(p10, p2) -> +(p2, p10)
+(p10, +(p2, x)) -> +(p2, +(p10, x))
+(p10, p5) -> +(p5, p10)
+(p10, +(p5, x)) -> +(p5, +(p10, x))

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

+(p1, p1) -> p2
+(p1, +(p2, p2)) -> p5
+(p1, +(p1, x)) -> +(p2, x)
+(p1, +(p2, +(p2, x))) -> +(p5, x)
+(p2, +(p2, p2)) -> +(p1, p5)
+(p2, +(p2, +(p2, x))) -> +(p1, +(p5, x))

where the Polynomial interpretation:
  POL(z)=  0  
  POL(p1)=  1  
  POL(p5)=  0  
  POL(p2)=  1  
  POL(p10)=  0  
  POL(y)=  1  
  POL(+(x1, x2))=  x1 + x2  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

+(p10, +(p2, x)) -> +(p2, +(p10, x))
+(p5, +(p2, x)) -> +(p2, +(p5, x))
+(p5, p2) -> +(p2, p5)
+(p10, p1) -> +(p1, p10)
+(p5, p1) -> +(p1, p5)
+(p10, +(p1, x)) -> +(p1, +(p10, x))
+(p10, p2) -> +(p2, p10)
+(p5, +(p1, x)) -> +(p1, +(p5, x))

where the Polynomial interpretation:
  POL(z)=  0  
  POL(p1)=  1  
  POL(p5)=  0  
  POL(p2)=  1  
  POL(p10)=  0  
  POL(y)=  2  
  POL(+(x1, x2))=  x1 + 2·x2  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

+(p10, p5) -> +(p5, p10)
+(p5, +(p5, x)) -> +(p10, x)
+(p10, +(p5, x)) -> +(p5, +(p10, x))
+(p5, p5) -> p10

where the Polynomial interpretation:
  POL(z)=  0  
  POL(p1)=  0  
  POL(p5)=  1  
  POL(p10)=  0  
  POL(p2)=  0  
  POL(y)=  1  
  POL(+(x1, x2))=  x1 + 2·x2  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →TRS4
                 ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

+(p2, +(p1, x)) -> +(p1, +(p2, x))
+(p2, p1) -> +(p1, p2)

where the Polynomial interpretation:
  POL(z)=  0  
  POL(p1)=  1  
  POL(p2)=  0  
  POL(y)=  0  
  POL(+(x1, x2))=  x1 + 2·x2  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →TRS5
                 ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

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

where the Polynomial interpretation:
  POL(z)=  0  
  POL(y)=  0  
  POL(+(x1, x2))=  1 + 2·x1 + x2  
was used.

All Rules of R can be deleted. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →TRS6
                 ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →TRS7
                 ↳Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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