Term Rewriting System R:

   [x]
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   bits(0) -> 0
   bits(s(x)) -> s(bits(half(s(x))))

Innermost Termination of R to be shown.


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

   bits(0) -> 0

where the Polynomial interpretation:
POL(bits(x_1)) = 1 + x_1
POL(s(x_1)) = x_1
POL(half(x_1)) = x_1
POL(0) = 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: 


   HALF(s(s(x))) -> HALF(x)
   BITS(s(x)) -> BITS(half(s(x)))
   BITS(s(x)) -> HALF(s(x))

Furthermore, R contains two SCCs.


SCC1:

HALF(s(s(x))) -> HALF(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(s(x_1)) = 2 + x_1
POL(HALF(x_1)) = 2 + x_1

The following Dependency Pairs can be deleted:

   HALF(s(s(x))) -> HALF(x)



 This transformation is resulting in no new subcycles.


SCC2:

BITS(s(x)) -> BITS(half(s(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(BITS(x_1)) = x_1
POL(s(x_1)) = 2*x_1
POL(half(x_1)) = x_1
POL(0) = 1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   half(s(0)) -> 0


 This transformation is resulting in one new subcycle:


SCC2.MRR1:

BITS(s(x)) -> BITS(half(s(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(BITS(x_1)) = 1 + x_1
POL(s(x_1)) = 1 + x_1
POL(half(x_1)) = x_1
POL(0) = 0

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   half(s(s(x))) -> s(half(x))


 This transformation is resulting in one new subcycle:


SCC2.MRR1.MRR1:

BITS(s(x)) -> BITS(half(s(x)))

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

   BITS(s(x)) -> BITS(half(s(x)))
no new Dependency Pairs
are created.

none
The transformation is resulting in no subcycles.




Innermost Termination of R successfully shown.


Duration: 0.854 seconds.