Term Rewriting System R:

   [x, y, z]
   +(x, 0) -> x
   +(minus(x), x) -> 0
   minus(0) -> 0
   minus(minus(x)) -> x
   minus(+(x, y)) -> +(minus(y), minus(x))
   *(x, 1) -> x
   *(x, 0) -> 0
   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(x, minus(y)) -> minus(*(x, y))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   *'(x, minus(y)) -> MINUS(*(x, y))
   *'(x, minus(y)) -> *'(x, y)
   *'(x, +(y, z)) -> +'(*(x, y), *(x, z))
   *'(x, +(y, z)) -> *'(x, y)
   *'(x, +(y, z)) -> *'(x, z)
   MINUS(+(x, y)) -> +'(minus(y), minus(x))
   MINUS(+(x, y)) -> MINUS(y)
   MINUS(+(x, y)) -> MINUS(x)

Furthermore, R contains two SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, y)
*'(x, minus(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(minus(x_1)) = 1 + x_1
POL(*'(x_1, x_2)) = 1 + x_1 + x_2
POL(+(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   *'(x, +(y, z)) -> *'(x, z)
   *'(x, +(y, z)) -> *'(x, y)
   *'(x, minus(y)) -> *'(x, y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.567 seconds.