Term Rewriting System R:

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


   LE(s(x), s(y)) -> LE(x, y)
   QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))
   QUOT(s(x), s(y)) -> MINUS(x, y)
   MINUS(s(x), y) -> IF_MINUS(le(s(x), y), s(x), y)
   MINUS(s(x), y) -> LE(s(x), y)
   IF_MINUS(false, s(x), 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 four 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:

IF_MINUS(false, s(x), y) -> MINUS(x, y)
MINUS(s(x), y) -> IF_MINUS(le(s(x), y), s(x), y)

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(le(x_1, x_2)) = 0
POL(true) = 0
POL(IF_MINUS(x_1, x_2, x_3)) = x_2
POL(MINUS(x_1, x_2)) = x_1
POL(0) = 0
POL(false) = 0

 resulting in no subcycles.


SCC3:

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), y) -> if_minus(le(s(x), y), s(x), y)
   minus(0, y) -> 0
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(le(x_1, x_2)) = 0
POL(minus(x_1, x_2)) = x_1
POL(true) = 0
POL(QUOT(x_1, x_2)) = x_1 + x_2
POL(if_minus(x_1, x_2, x_3)) = x_2
POL(0) = 0
POL(false) = 0

 resulting in no subcycles.


SCC4:

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(s(x), y) -> if_minus(le(s(x), y), s(x), y)
   minus(0, y) -> 0
   if_minus(true, s(x), y) -> 0
   if_minus(false, s(x), y) -> s(minus(x, y))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(le(x_1, x_2)) = 0
POL(minus(x_1, x_2)) = x_1
POL(true) = 0
POL(if_minus(x_1, x_2, x_3)) = x_2
POL(0) = 0
POL(false) = 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: 0.981 seconds.