Term Rewriting System R:

   [x, y]
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(quot(x, s(s(0))))))

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: 


   QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))
   QUOT(s(x), s(y)) -> MINUS(x, y)
   MINUS(x, s(y)) -> PRED(minus(x, y))
   MINUS(x, s(y)) -> MINUS(x, y)
   LOG(s(s(x))) -> LOG(s(quot(x, s(s(0)))))
   LOG(s(s(x))) -> QUOT(x, s(s(0)))

Furthermore, R contains three SCCs.


SCC1:

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



 This transformation is resulting in no new subcycles.


SCC2:

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


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

 resulting in no subcycles.


SCC3:

LOG(s(s(x))) -> LOG(s(quot(x, s(s(0)))))

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

Additionally, the following rules can be oriented: 

   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   pred(s(x)) -> x


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

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.723 seconds.