Term Rewriting System R:

   [x, y]
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(s(x), s(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: 


   LE(s(x), s(y)) -> LE(x, y)
   QUOT(s(x), s(y)) -> QUOT(minus(s(x), s(y)), s(y))
   QUOT(s(x), s(y)) -> MINUS(s(x), s(y))
   MINUS(s(x), s(y)) -> MINUS(x, y)

Furthermore, R contains three SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

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.


SCC3:

QUOT(s(x), s(y)) -> QUOT(minus(s(x), s(y)), s(y))

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

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


   QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))

The transformation is resulting in one subcycle:


SCC3.Rewr1:

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


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

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.633 seconds.