Term Rewriting System R:

   [x, y]
   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   div(x, 0) -> 0
   div(0, y) -> 0
   div(s(x), s(y)) -> if(lt(x, y), 0, s(div(-(x, y), s(y))))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   LT(s(x), s(y)) -> LT(x, y)
   DIV(s(x), s(y)) -> IF(lt(x, y), 0, s(div(-(x, y), s(y))))
   DIV(s(x), s(y)) -> LT(x, y)
   DIV(s(x), s(y)) -> DIV(-(x, y), s(y))
   DIV(s(x), s(y)) -> -'(x, y)
   -'(s(x), s(y)) -> -'(x, y)

Furthermore, R contains three SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

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

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


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

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.727 seconds.