Term Rewriting System R:
[X, Y]
f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Termination of R to be shown.



   R
Removing Redundant Rules



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

f(X) -> nf(X)

where the Polynomial interpretation:
  POL(n__f(x1))=  1 + x1  
  POL(n__true)=  0  
  POL(activate(x1))=  2·x1  
  POL(if(x1, x2, x3))=  x1 + x2 + 2·x3  
  POL(c)=  0  
  POL(false)=  0  
  POL(true)=  0  
  POL(f(x1))=  2 + x1  
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:

if(false, X, Y) -> activate(Y)

where the Polynomial interpretation:
  POL(activate(x1))=  x1  
  POL(n__true)=  0  
  POL(n__f(x1))=  x1  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(c)=  0  
  POL(false)=  1  
  POL(true)=  0  
  POL(f(x1))=  x1  
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:

true -> ntrue
if(true, X, Y) -> X
activate(X) -> X

where the Polynomial interpretation:
  POL(n__f(x1))=  x1  
  POL(n__true)=  0  
  POL(activate(x1))=  1 + x1  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(c)=  0  
  POL(true)=  1  
  POL(f(x1))=  x1  
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:

f(X) -> if(X, c, nf(ntrue))

where the Polynomial interpretation:
  POL(activate(x1))=  2·x1  
  POL(n__true)=  0  
  POL(n__f(x1))=  1 + x1  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(c)=  0  
  POL(true)=  0  
  POL(f(x1))=  2 + x1  
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
             ...
               →TRS5
Removing Redundant Rules



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

activate(ntrue) -> true

where the Polynomial interpretation:
  POL(activate(x1))=  x1  
  POL(n__f(x1))=  x1  
  POL(n__true)=  1  
  POL(true)=  0  
  POL(f(x1))=  x1  
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
             ...
               →TRS6
Removing Redundant Rules



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

activate(nf(X)) -> f(activate(X))

where the Polynomial interpretation:
  POL(activate(x1))=  x1  
  POL(n__f(x1))=  1 + x1  
  POL(f(x1))=  x1  
was used.

All Rules of R can be deleted.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS7
Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS8
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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