Term Rewriting System R:

   [X, Y]
   minus(X, 0) -> X
   minus(s(X), s(Y)) -> p(minus(X, Y))
   p(s(X)) -> X
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> s(div(minus(X, Y), s(Y)))

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: 


   DIV(s(X), s(Y)) -> DIV(minus(X, Y), s(Y))
   DIV(s(X), s(Y)) -> MINUS(X, Y)
   MINUS(s(X), s(Y)) -> P(minus(X, Y))
   MINUS(s(X), s(Y)) -> MINUS(X, Y)

Furthermore, R contains two SCCs.


SCC1:

MINUS(s(X), s(Y)) -> MINUS(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(MINUS(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   MINUS(s(X), s(Y)) -> MINUS(X, Y)



 This transformation is resulting in no new subcycles.


SCC2:

DIV(s(X), s(Y)) -> DIV(minus(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: 

   minus(X, 0) -> X
   minus(s(X), s(Y)) -> p(minus(X, Y))
   p(s(X)) -> X


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

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.578 seconds.