Term Rewriting System R:

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

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: 


   +'(x, s(y)) -> +'(x, y)
   SUM(s(x)) -> +'(sum(x), s(x))
   SUM(s(x)) -> SUM(x)

Furthermore, R contains two SCCs.


SCC1:

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

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(+'(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 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: 0.492 seconds.