Term Rewriting System R:

   [y, x]
   leq(0, y) -> true
   leq(s(x), 0) -> false
   leq(s(x), s(y)) -> leq(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   MOD(s(x), s(y)) -> IF(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))
   MOD(s(x), s(y)) -> LEQ(y, x)
   MOD(s(x), s(y)) -> MOD(-(s(x), s(y)), s(y))
   MOD(s(x), s(y)) -> -'(s(x), s(y))
   -'(s(x), s(y)) -> -'(x, y)
   LEQ(s(x), s(y)) -> LEQ(x, y)

Furthermore, R contains three SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

   LEQ(s(x), s(y)) -> LEQ(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:

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

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

   MOD(s(x), s(y)) -> MOD(-(s(x), s(y)), s(y))
one new Dependency Pair
is created:


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

The transformation is resulting in one subcycle:


SCC3.Rewr1:

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

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   -(s(x), s(y)) -> -(x, y)
   -(x, 0) -> x


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(-(x_1, x_2)) = x_1
POL(MOD(x_1, x_2)) = x_1 + x_2
POL(0) = 1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.731 seconds.