Term Rewriting System R:

   [y, x]
   -(0, y) -> 0
   -(x, 0) -> x
   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   p(0) -> 0
   p(s(x)) -> x

Termination of R to be shown.


R contains the following Dependency Pairs: 


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

Furthermore, R contains one SCC.


SCC1:

-'(x, s(y)) -> -'(x, p(s(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)) = x_1
POL(-'(x_1, x_2)) = x_1 + x_2
POL(0) = 0
POL(p(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
   -(0, y) -> 0
   -(x, 0) -> x


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

-'(x, s(y)) -> -'(x, p(s(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(0) = 0
POL(p(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   p(s(x)) -> x


 This transformation is resulting in one new subcycle:


SCC1.MRR1.MRR1:

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

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC1.MRR1.MRR1.NOC1:

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

On this Scc, a Narrowing SCC transformation can be performed.
 
As a result of transforming the rule 

   -'(x, s(y)) -> -'(x, p(s(y)))
no new Dependency Pairs
are created.

none
The transformation is resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.552 seconds.