Term Rewriting System R:
[x, y]
not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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

not(true) -> false
not(false) -> true

where the Polynomial interpretation:
  POL(0)=  0  
  POL(odd(x1))=  x1  
  POL(false)=  0  
  POL(true)=  0  
  POL(s(x1))=  1 + x1  
  POL(not(x1))=  1 + x1  
  POL(+(x1, x2))=  x1 + x2  
was used.

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


   R
RRRPolo
       →TRS2
Removing Redundant Rules



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

+(x, 0) -> x
odd(0) -> false

where the Polynomial interpretation:
  POL(0)=  1  
  POL(odd(x1))=  x1  
  POL(false)=  0  
  POL(s(x1))=  x1  
  POL(not(x1))=  x1  
  POL(+(x1, x2))=  x1 + x2  
was used.

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


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
Removing Redundant Rules



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

+(s(x), y) -> s(+(x, y))
odd(s(x)) -> not(odd(x))

where the Polynomial interpretation:
  POL(odd(x1))=  x1  
  POL(s(x1))=  1 + x1  
  POL(not(x1))=  x1  
  POL(+(x1, x2))=  2·x1 + x2  
was used.

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


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS4
Removing Redundant Rules



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

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

where the Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(+(x1, x2))=  x1 + 2·x2  
was used.

All Rules of R can be deleted.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS5
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes