Term Rewriting System R:

   [x, y]
   times(x, 0) -> 0
   times(x, s(y)) -> plus(times(x, y), x)
   plus(x, 0) -> x
   plus(0, x) -> x
   plus(x, s(y)) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(x, y))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   TIMES(x, s(y)) -> PLUS(times(x, y), x)
   TIMES(x, s(y)) -> TIMES(x, y)
   PLUS(s(x), y) -> PLUS(x, y)
   PLUS(x, s(y)) -> PLUS(x, y)

Furthermore, R contains two SCCs.


SCC1:

PLUS(x, s(y)) -> PLUS(x, y)
PLUS(s(x), y) -> PLUS(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(PLUS(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   PLUS(x, s(y)) -> PLUS(x, y)
   PLUS(s(x), y) -> PLUS(x, y)



 This transformation is resulting in no new subcycles.


SCC2:

TIMES(x, s(y)) -> TIMES(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(TIMES(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   TIMES(x, s(y)) -> TIMES(x, y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.506 seconds.