Term Rewriting System R:

   [x]
   sum(0) -> 0
   sum(s(x)) -> +(sum(x), s(x))
   sum1(0) -> 0
   sum1(s(x)) -> s(+(sum1(x), +(x, x)))

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   SUM1(s(x)) -> SUM1(x)
   SUM(s(x)) -> SUM(x)

Furthermore, R contains two SCCs.


SCC1:

SUM1(s(x)) -> SUM1(x)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(SUM1(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   SUM1(s(x)) -> SUM1(x)



 This transformation is resulting in no new subcycles.


SCC2:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(SUM(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.480 seconds.