Term Rewriting System R:

   [x, y]
   sqr(0) -> 0
   sqr(s(x)) -> +(sqr(x), s(double(x)))
   sqr(s(x)) -> s(+(sqr(x), double(x)))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   +'(x, s(y)) -> +'(x, y)
   DOUBLE(s(x)) -> DOUBLE(x)
   SQR(s(x)) -> +'(sqr(x), s(double(x)))
   SQR(s(x)) -> SQR(x)
   SQR(s(x)) -> DOUBLE(x)
   SQR(s(x)) -> +'(sqr(x), double(x))

Furthermore, R contains three SCCs.


SCC1:

+'(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:

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

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

The following Dependency Pairs can be deleted:

   SQR(s(x)) -> SQR(x)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.557 seconds.