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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

SUM(s(x)) -> SQR(s(x))
SUM(s(x)) -> SUM(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pair:

SUM(s(x)) -> SUM(x)


Rules:


sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)





The following dependency pair can be strictly oriented:

SUM(s(x)) -> SUM(x)


Additionally, the following rules can be oriented:

sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(SUM(x1))=  x1  
  POL(sqr(x1))=  0  
  POL(*(x1, x2))=  0  
  POL(sum(x1))=  0  
  POL(s(x1))=  1 + x1  
  POL(+(x1, x2))=  0  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)





Using the Dependency Graph resulted in no new DP problems.

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