Term Rewriting System R:

   [x, y, z]
   0(#) -> #
   +(#, x) -> x
   +(x, #) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
   +(+(x, y), z) -> +(x, +(y, z))
   -(#, x) -> #
   -(x, #) -> x
   -(0(x), 0(y)) -> 0(-(x, y))
   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
   -(1(x), 0(y)) -> 1(-(x, y))
   -(1(x), 1(y)) -> 0(-(x, y))
   not(true) -> false
   not(false) -> true
   if(true, x, y) -> x
   if(false, x, y) -> y
   ge(0(x), 0(y)) -> ge(x, y)
   ge(0(x), 1(y)) -> not(ge(y, x))
   ge(1(x), 0(y)) -> ge(x, y)
   ge(1(x), 1(y)) -> ge(x, y)
   ge(x, #) -> true
   ge(#, 0(x)) -> ge(#, x)
   ge(#, 1(x)) -> false
   log(x) -> -(log'(x), 1(#))
   log'(#) -> #
   log'(1(x)) -> +(log'(x), 1(#))
   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   log(x) -> -(log'(x), 1(#))

where the Polynomial interpretation:
POL(log(x_1)) = 1 + x_1
POL(1(x_1)) = 2*x_1
POL(not(x_1)) = x_1
POL(-(x_1, x_2)) = x_1 + x_2
POL(true) = 0
POL(ge(x_1, x_2)) = x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0(x_1)) = 2*x_1
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(log'(x_1)) = x_1
POL(#) = 0
POL(false) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   log'(#) -> #

where the Polynomial interpretation:
POL(1(x_1)) = 2*x_1
POL(not(x_1)) = x_1
POL(-(x_1, x_2)) = x_1 + x_2
POL(true) = 0
POL(ge(x_1, x_2)) = x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0(x_1)) = 2*x_1
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(log'(x_1)) = 1 + x_1
POL(#) = 0
POL(false) = 0
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


R contains the following Dependency Pairs: 


   GE(0(x), 0(y)) -> GE(x, y)
   GE(0(x), 1(y)) -> NOT(ge(y, x))
   GE(0(x), 1(y)) -> GE(y, x)
   GE(1(x), 1(y)) -> GE(x, y)
   GE(#, 0(x)) -> GE(#, x)
   GE(1(x), 0(y)) -> GE(x, y)
   -'(0(x), 1(y)) -> -'(-(x, y), 1(#))
   -'(0(x), 1(y)) -> -'(x, y)
   -'(0(x), 0(y)) -> 0'(-(x, y))
   -'(0(x), 0(y)) -> -'(x, y)
   -'(1(x), 1(y)) -> 0'(-(x, y))
   -'(1(x), 1(y)) -> -'(x, y)
   -'(1(x), 0(y)) -> -'(x, y)
   +'(0(x), 1(y)) -> +'(x, y)
   +'(1(x), 0(y)) -> +'(x, y)
   +'(0(x), 0(y)) -> 0'(+(x, y))
   +'(0(x), 0(y)) -> +'(x, y)
   +'(1(x), 1(y)) -> 0'(+(+(x, y), 1(#)))
   +'(1(x), 1(y)) -> +'(+(x, y), 1(#))
   +'(1(x), 1(y)) -> +'(x, y)
   +'(+(x, y), z) -> +'(x, +(y, z))
   +'(+(x, y), z) -> +'(y, z)
   LOG'(0(x)) -> IF(ge(x, 1(#)), +(log'(x), 1(#)), #)
   LOG'(0(x)) -> GE(x, 1(#))
   LOG'(0(x)) -> +'(log'(x), 1(#))
   LOG'(0(x)) -> LOG'(x)
   LOG'(1(x)) -> +'(log'(x), 1(#))
   LOG'(1(x)) -> LOG'(x)

Furthermore, R contains five SCCs.


SCC1:

GE(#, 0(x)) -> GE(#, x)

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(GE(x_1, x_2)) = 1 + x_1 + x_2
POL(0(x_1)) = 1 + x_1
POL(#) = 1

The following Dependency Pairs can be deleted:

   GE(#, 0(x)) -> GE(#, x)



 This transformation is resulting in no new subcycles.


SCC2:

GE(1(x), 0(y)) -> GE(x, y)
GE(1(x), 1(y)) -> GE(x, y)
GE(0(x), 1(y)) -> GE(y, x)
GE(0(x), 0(y)) -> GE(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(1(x_1)) = 1 + x_1
POL(GE(x_1, x_2)) = 1 + x_1 + x_2
POL(0(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   GE(1(x), 0(y)) -> GE(x, y)
   GE(1(x), 1(y)) -> GE(x, y)
   GE(0(x), 1(y)) -> GE(y, x)
   GE(0(x), 0(y)) -> GE(x, y)



 This transformation is resulting in no new subcycles.


SCC3:

-'(1(x), 0(y)) -> -'(x, y)
-'(1(x), 1(y)) -> -'(x, y)
-'(0(x), 0(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(-(x, y), 1(#))

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(1(x_1)) = x_1
POL(-(x_1, x_2)) = x_1 + x_2
POL(-'(x_1, x_2)) = 1 + x_1 + x_2
POL(0(x_1)) = x_1
POL(#) = 0

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   ge(0(x), 0(y)) -> ge(x, y)
   ge(0(x), 1(y)) -> not(ge(y, x))
   ge(1(x), 1(y)) -> ge(x, y)
   ge(x, #) -> true
   ge(#, 0(x)) -> ge(#, x)
   ge(1(x), 0(y)) -> ge(x, y)
   ge(#, 1(x)) -> false
   not(true) -> false
   not(false) -> true
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(0(x), 0(y)) -> 0(+(x, y))
   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
   +(+(x, y), z) -> +(x, +(y, z))
   +(x, #) -> x
   +(#, x) -> x
   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
   log'(1(x)) -> +(log'(x), 1(#))
   if(false, x, y) -> y
   if(true, x, y) -> x


 This transformation is resulting in one new subcycle:


SCC3.MRR1:

-'(1(x), 1(y)) -> -'(x, y)
-'(0(x), 0(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(x, y)
-'(0(x), 1(y)) -> -'(-(x, y), 1(#))
-'(1(x), 0(y)) -> -'(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(1(x_1)) = 1 + x_1
POL(-(x_1, x_2)) = x_1 + x_2
POL(-'(x_1, x_2)) = 1 + x_1 + x_2
POL(0(x_1)) = 1 + x_1
POL(#) = 0

The following Dependency Pairs can be deleted:

   -'(1(x), 1(y)) -> -'(x, y)
   -'(0(x), 0(y)) -> -'(x, y)
   -'(0(x), 1(y)) -> -'(x, y)
   -'(0(x), 1(y)) -> -'(-(x, y), 1(#))
   -'(1(x), 0(y)) -> -'(x, y)



 This transformation is resulting in no new subcycles.


SCC4:

+'(+(x, y), z) -> +'(y, z)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(0(x), 0(y)) -> +'(x, y)
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(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(1(x_1)) = x_1
POL(+'(x_1, x_2)) = 1 + x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0(x_1)) = x_1
POL(#) = 0

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   ge(0(x), 0(y)) -> ge(x, y)
   ge(0(x), 1(y)) -> not(ge(y, x))
   ge(1(x), 1(y)) -> ge(x, y)
   ge(x, #) -> true
   ge(#, 0(x)) -> ge(#, x)
   ge(1(x), 0(y)) -> ge(x, y)
   ge(#, 1(x)) -> false
   not(true) -> false
   not(false) -> true
   -(0(x), 1(y)) -> 1(-(-(x, y), 1(#)))
   -(0(x), 0(y)) -> 0(-(x, y))
   -(1(x), 1(y)) -> 0(-(x, y))
   -(1(x), 0(y)) -> 1(-(x, y))
   -(x, #) -> x
   -(#, x) -> #
   log'(0(x)) -> if(ge(x, 1(#)), +(log'(x), 1(#)), #)
   log'(1(x)) -> +(log'(x), 1(#))
   if(false, x, y) -> y
   if(true, x, y) -> x


 This transformation is resulting in one new subcycle:


SCC4.MRR1:

+'(+(x, y), z) -> +'(x, +(y, z))
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(0(x), 0(y)) -> +'(x, y)
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(+(x, y), z) -> +'(y, z)

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(1(x_1)) = 1 + x_1
POL(+'(x_1, x_2)) = 1 + x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0(x_1)) = x_1
POL(#) = 0

The following Dependency Pairs can be deleted:

   +'(1(x), 1(y)) -> +'(x, y)
   +'(1(x), 1(y)) -> +'(+(x, y), 1(#))
   +'(1(x), 0(y)) -> +'(x, y)
   +'(0(x), 1(y)) -> +'(x, y)

The following rules of R can be deleted: 

   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))


 This transformation is resulting in one new subcycle:


SCC4.MRR1.MRR1:

+'(+(x, y), z) -> +'(y, z)
+'(0(x), 0(y)) -> +'(x, y)
+'(+(x, y), z) -> +'(x, +(y, z))

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(1(x_1)) = x_1
POL(+'(x_1, x_2)) = x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0(x_1)) = 1 + x_1
POL(#) = 0

The following Dependency Pairs can be deleted:

   +'(0(x), 0(y)) -> +'(x, y)

The following rules of R can be deleted: 

   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(0(x), 0(y)) -> 0(+(x, y))
   0(#) -> #
   +(x, #) -> x
   +(#, x) -> x


 This transformation is resulting in one new subcycle:


SCC4.MRR1.MRR1.MRR1:

+'(+(x, y), z) -> +'(x, +(y, z))
+'(+(x, y), z) -> +'(y, z)

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(+'(x_1, x_2)) = 1 + x_1 + x_2
POL(+(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   +'(+(x, y), z) -> +'(y, z)

No rules of R can be deleted.


 This transformation is resulting in one new subcycle:


SCC4.MRR1.MRR1.MRR1.MRR1:

+'(+(x, y), z) -> +'(x, +(y, z))

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(+(x_1, x_2)) = 1 + x_1
POL(+'(x_1, x_2)) = x_1

 resulting in no subcycles.


SCC5:

LOG'(1(x)) -> LOG'(x)
LOG'(0(x)) -> LOG'(x)

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(1(x_1)) = 1 + x_1
POL(LOG'(x_1)) = 1 + x_1
POL(0(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   LOG'(1(x)) -> LOG'(x)
   LOG'(0(x)) -> LOG'(x)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 9.327 seconds.