Term Rewriting System R:

   [X, Y]
   min(X, 0) -> X
   min(s(X), s(Y)) -> min(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(min(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(min(X, Y), s(Y))
   QUOT(s(X), s(Y)) -> MIN(X, Y)
   LOG(s(s(X))) -> LOG(s(quot(X, s(s(0)))))
   LOG(s(s(X))) -> QUOT(X, s(s(0)))
   MIN(s(X), s(Y)) -> MIN(X, Y)

Furthermore, R contains three SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

QUOT(s(X), s(Y)) -> QUOT(min(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: 

   min(s(X), s(Y)) -> min(X, Y)
   min(X, 0) -> X


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(QUOT(x_1, x_2)) = x_1 + x_2
POL(min(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(min(X, Y), s(Y)))
   min(s(X), s(Y)) -> min(X, Y)
   min(X, 0) -> X


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(min(x_1, x_2)) = 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: 0.691 seconds.