Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


   EXP(x, s(y)) -> *'(x, exp(x, y))
   EXP(x, s(y)) -> EXP(x, y)
   -'(s(x), s(y)) -> -'(x, y)
   *'(s(x), y) -> *'(x, y)

Furthermore, R contains three SCCs.


SCC1:

*'(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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

-'(s(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:

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



 This transformation is resulting in no new subcycles.


SCC3:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.528 seconds.