Term Rewriting System R:

   [x, y]
   p(s(x)) -> x
   s(p(x)) -> x
   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   +(p(x), y) -> p(+(x, y))
   minus(0) -> 0
   minus(s(x)) -> p(minus(x))
   minus(p(x)) -> s(minus(x))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   *(p(x), y) -> +(*(x, y), minus(y))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   +'(s(x), y) -> S(+(x, y))
   +'(s(x), y) -> +'(x, y)
   +'(p(x), y) -> P(+(x, y))
   +'(p(x), y) -> +'(x, y)
   MINUS(p(x)) -> S(minus(x))
   MINUS(p(x)) -> MINUS(x)
   MINUS(s(x)) -> P(minus(x))
   MINUS(s(x)) -> MINUS(x)
   *'(s(x), y) -> +'(*(x, y), y)
   *'(s(x), y) -> *'(x, y)
   *'(p(x), y) -> +'(*(x, y), minus(y))
   *'(p(x), y) -> *'(x, y)
   *'(p(x), y) -> MINUS(y)

Furthermore, R contains three SCCs.


SCC1:

+'(p(x), y) -> +'(x, y)
+'(s(x), 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
POL(p(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

MINUS(s(x)) -> MINUS(x)
MINUS(p(x)) -> MINUS(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(MINUS(x_1)) = 1 + x_1
POL(p(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   MINUS(s(x)) -> MINUS(x)
   MINUS(p(x)) -> MINUS(x)



 This transformation is resulting in no new subcycles.


SCC3:

*'(p(x), y) -> *'(x, y)
*'(s(x), 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
POL(p(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   *'(p(x), y) -> *'(x, y)
   *'(s(x), y) -> *'(x, y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.653 seconds.