Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


   PLUS(s(x), y) -> PLUS(x, s(y))
   PLUS(s(x), y) -> PLUS(x, y)
   PLUS(s(x), y) -> PLUS(minus(x, y), double(y))
   PLUS(s(x), y) -> MINUS(x, y)
   PLUS(s(x), y) -> DOUBLE(y)
   MINUS(s(x), s(y)) -> MINUS(x, y)
   DOUBLE(s(x)) -> DOUBLE(x)

Furthermore, R contains three SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

DOUBLE(s(x)) -> DOUBLE(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(DOUBLE(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   DOUBLE(s(x)) -> DOUBLE(x)



 This transformation is resulting in no new subcycles.


SCC3:

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

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   minus(s(x), s(y)) -> minus(x, y)
   minus(x, 0) -> x


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(PLUS(x_1, x_2)) = x_1
POL(minus(x_1, x_2)) = x_1
POL(double(x_1)) = 0
POL(0) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.625 seconds.