Term Rewriting System R:
[X, Y, X1, X2, X3]
af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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

af(X) -> f(X)

where the Polynomial interpretation:
  POL(c)=  0  
  POL(if(x1, x2, x3))=  x1 + 2·x2 + x3  
  POL(false)=  0  
  POL(a__if(x1, x2, x3))=  x1 + 2·x2 + 2·x3  
  POL(true)=  0  
  POL(mark(x1))=  2·x1  
  POL(f(x1))=  1 + 2·x1  
  POL(a__f(x1))=  2 + 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:

mark(false) -> false
aif(false, X, Y) -> mark(Y)

where the Polynomial interpretation:
  POL(if(x1, x2, x3))=  x1 + 2·x2 + x3  
  POL(c)=  0  
  POL(false)=  1  
  POL(a__if(x1, x2, x3))=  x1 + 2·x2 + 2·x3  
  POL(true)=  0  
  POL(mark(x1))=  2·x1  
  POL(f(x1))=  2·x1  
  POL(a__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
Removing Redundant Rules



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

af(X) -> aif(mark(X), c, f(true))

where the Polynomial interpretation:
  POL(if(x1, x2, x3))=  x1 + 2·x2 + x3  
  POL(c)=  0  
  POL(a__if(x1, x2, x3))=  x1 + 2·x2 + x3  
  POL(true)=  0  
  POL(mark(x1))=  2·x1  
  POL(f(x1))=  1 + 2·x1  
  POL(a__f(x1))=  2 + 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
             ...
               →TRS4
Removing Redundant Rules



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

aif(X1, X2, X3) -> if(X1, X2, X3)
aif(true, X, Y) -> mark(X)

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



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

mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)

where the Polynomial interpretation:
  POL(c)=  0  
  POL(if(x1, x2, x3))=  1 + x1 + x2 + x3  
  POL(a__if(x1, x2, x3))=  x1 + x2 + x3  
  POL(true)=  0  
  POL(mark(x1))=  x1  
  POL(f(x1))=  x1  
  POL(a__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:

mark(f(X)) -> af(mark(X))

where the Polynomial interpretation:
  POL(c)=  0  
  POL(true)=  0  
  POL(mark(x1))=  x1  
  POL(f(x1))=  1 + x1  
  POL(a__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
             ...
               →TRS7
Removing Redundant Rules



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

mark(true) -> true
mark(c) -> c

where the Polynomial interpretation:
  POL(c)=  0  
  POL(true)=  0  
  POL(mark(x1))=  1 + x1  
was used.

All Rules of R can be deleted.


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



R contains no Dependency Pairs and therefore no SCCs.

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