Term Rewriting System R:

   [X, Y, X1, X2, X3]
   active(f(X)) -> mark(if(X, c, f(true)))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(f(X)) -> f(active(X))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
   f(mark(X)) -> mark(f(X))
   f(ok(X)) -> ok(f(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   proper(f(X)) -> f(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(c) -> ok(c)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Termination of R to be shown.


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

   active(if(false, X, Y)) -> mark(Y)

where the Polynomial interpretation:
POL(active(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(top(x_1)) = x_1
POL(true) = 0
POL(mark(x_1)) = x_1
POL(ok(x_1)) = x_1
POL(f(x_1)) = x_1
POL(c) = 0
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(false) = 1
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: 


   ACTIVE(if(X1, X2, X3)) -> IF(active(X1), X2, X3)
   ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)
   ACTIVE(if(X1, X2, X3)) -> IF(X1, active(X2), X3)
   ACTIVE(if(X1, X2, X3)) -> ACTIVE(X2)
   ACTIVE(f(X)) -> IF(X, c, f(true))
   ACTIVE(f(X)) -> F(true)
   ACTIVE(f(X)) -> F(active(X))
   ACTIVE(f(X)) -> ACTIVE(X)
   IF(mark(X1), X2, X3) -> IF(X1, X2, X3)
   IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
   IF(X1, mark(X2), X3) -> IF(X1, X2, X3)
   F(mark(X)) -> F(X)
   F(ok(X)) -> F(X)
   PROPER(f(X)) -> F(proper(X))
   PROPER(f(X)) -> PROPER(X)
   PROPER(if(X1, X2, X3)) -> IF(proper(X1), proper(X2), proper(X3))
   PROPER(if(X1, X2, X3)) -> PROPER(X1)
   PROPER(if(X1, X2, X3)) -> PROPER(X2)
   PROPER(if(X1, X2, X3)) -> PROPER(X3)
   TOP(mark(X)) -> TOP(proper(X))
   TOP(mark(X)) -> PROPER(X)
   TOP(ok(X)) -> TOP(active(X))
   TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains five SCCs.


SCC1:

IF(X1, mark(X2), X3) -> IF(X1, X2, X3)
IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
IF(mark(X1), X2, X3) -> IF(X1, X2, X3)

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(mark(x_1)) = 1 + x_1
POL(ok(x_1)) = 1 + x_1
POL(IF(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3

The following Dependency Pairs can be deleted:

   IF(X1, mark(X2), X3) -> IF(X1, X2, X3)
   IF(ok(X1), ok(X2), ok(X3)) -> IF(X1, X2, X3)
   IF(mark(X1), X2, X3) -> IF(X1, X2, X3)



 This transformation is resulting in no new subcycles.


SCC2:

F(ok(X)) -> F(X)
F(mark(X)) -> F(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(mark(x_1)) = 1 + x_1
POL(F(x_1)) = 1 + x_1
POL(ok(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   F(ok(X)) -> F(X)
   F(mark(X)) -> F(X)



 This transformation is resulting in no new subcycles.


SCC3:

ACTIVE(f(X)) -> ACTIVE(X)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X2)
ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)

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(ACTIVE(x_1)) = 1 + x_1
POL(f(x_1)) = 1 + x_1
POL(if(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3

The following Dependency Pairs can be deleted:

   ACTIVE(f(X)) -> ACTIVE(X)
   ACTIVE(if(X1, X2, X3)) -> ACTIVE(X2)
   ACTIVE(if(X1, X2, X3)) -> ACTIVE(X1)



 This transformation is resulting in no new subcycles.


SCC4:

PROPER(if(X1, X2, X3)) -> PROPER(X3)
PROPER(if(X1, X2, X3)) -> PROPER(X2)
PROPER(if(X1, X2, X3)) -> PROPER(X1)
PROPER(f(X)) -> PROPER(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(PROPER(x_1)) = 1 + x_1
POL(f(x_1)) = 1 + x_1
POL(if(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3

The following Dependency Pairs can be deleted:

   PROPER(if(X1, X2, X3)) -> PROPER(X3)
   PROPER(if(X1, X2, X3)) -> PROPER(X2)
   PROPER(if(X1, X2, X3)) -> PROPER(X1)
   PROPER(f(X)) -> PROPER(X)



 This transformation is resulting in no new subcycles.


SCC5:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(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(active(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(TOP(x_1)) = x_1
POL(true) = 0
POL(mark(x_1)) = x_1
POL(ok(x_1)) = x_1
POL(f(x_1)) = x_1
POL(c) = 0
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(false) = 0

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))


 This transformation is resulting in one new subcycle:


SCC5.MRR1:

TOP(mark(X)) -> TOP(proper(X))
TOP(ok(X)) -> TOP(active(X))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   proper(c) -> ok(c)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(f(X)) -> f(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(if(X1, X2, X3)) -> if(X1, active(X2), X3)
   active(f(X)) -> mark(if(X, c, f(true)))
   active(f(X)) -> f(active(X))
   active(if(true, X, Y)) -> mark(X)
   f(mark(X)) -> mark(f(X))
   f(ok(X)) -> ok(f(X))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   if(X1, mark(X2), X3) -> mark(if(X1, X2, X3))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(active(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(true) = 1
POL(TOP(x_1)) = x_1
POL(mark(x_1)) = 1 + x_1
POL(ok(x_1)) = x_1
POL(c) = 0
POL(f(x_1)) = 1 + x_1
POL(if(x_1, x_2, x_3)) = x_1 + x_2
POL(false) = 0

 resulting in one subcycle.


SCC5.MRR1.Polo1:

TOP(ok(X)) -> TOP(active(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(active(x_1)) = x_1
POL(TOP(x_1)) = 1 + x_1
POL(true) = 0
POL(mark(x_1)) = x_1
POL(ok(x_1)) = x_1
POL(f(x_1)) = x_1
POL(c) = 0
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   proper(c) -> ok(c)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(f(X)) -> f(proper(X))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))


 This transformation is resulting in one new subcycle:


SCC5.MRR1.Polo1.MRR1:

TOP(ok(X)) -> TOP(active(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(active(x_1)) = x_1
POL(TOP(x_1)) = 1 + x_1
POL(true) = 0
POL(mark(x_1)) = x_1
POL(ok(x_1)) = 1 + x_1
POL(f(x_1)) = x_1
POL(c) = 0
POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3

The following Dependency Pairs can be deleted:

   TOP(ok(X)) -> TOP(active(X))



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 1.889 seconds.