Term Rewriting System R:

   [x, y]
   active(f(x)) -> mark(x)
   active(f(x)) -> f(active(x))
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   top(found(x)) -> top(active(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   f(found(x)) -> found(f(x))
   f(mark(x)) -> mark(f(x))
   start(ok(x)) -> found(x)

Termination of R to be shown.


R contains the following Dependency Pairs: 


   F(found(x)) -> F(x)
   F(ok(x)) -> F(x)
   F(mark(x)) -> F(x)
   PROPER(f(x)) -> F(proper(x))
   PROPER(f(x)) -> PROPER(x)
   TOP(mark(x)) -> TOP(check(x))
   TOP(mark(x)) -> CHECK(x)
   TOP(active(c)) -> TOP(mark(c))
   TOP(found(x)) -> TOP(active(x))
   TOP(found(x)) -> ACTIVE(x)
   CHECK(x) -> START(match(f(X), x))
   CHECK(x) -> MATCH(f(X), x)
   CHECK(x) -> F(X)
   CHECK(f(x)) -> F(check(x))
   CHECK(f(x)) -> CHECK(x)
   MATCH(f(x), f(y)) -> F(match(x, y))
   MATCH(f(x), f(y)) -> MATCH(x, y)
   MATCH(X, x) -> PROPER(x)
   ACTIVE(f(x)) -> F(active(x))
   ACTIVE(f(x)) -> ACTIVE(x)

Furthermore, R contains six SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

   F(mark(x)) -> F(x)
   F(ok(x)) -> F(x)
   F(found(x)) -> F(x)



 This transformation is resulting in no new subcycles.


SCC2:

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

The following Dependency Pairs can be deleted:

   PROPER(f(x)) -> PROPER(x)



 This transformation is resulting in no new subcycles.


SCC3:

ACTIVE(f(x)) -> 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)) = 1 + x_1
POL(f(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   ACTIVE(f(x)) -> ACTIVE(x)



 This transformation is resulting in no new subcycles.


SCC4:

MATCH(f(x), f(y)) -> MATCH(x, y)

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(f(x_1)) = 1 + x_1
POL(MATCH(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   MATCH(f(x), f(y)) -> MATCH(x, y)



 This transformation is resulting in no new subcycles.


SCC5:

CHECK(f(x)) -> CHECK(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(CHECK(x_1)) = 1 + x_1
POL(f(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   CHECK(f(x)) -> CHECK(x)



 This transformation is resulting in no new subcycles.


SCC6:

TOP(active(c)) -> TOP(mark(c))
TOP(found(x)) -> TOP(active(x))
TOP(mark(x)) -> TOP(check(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(start(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(X) = 0
POL(TOP(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(match(x_1, x_2)) = x_1 + x_2
POL(check(x_1)) = x_1
POL(ok(x_1)) = x_1
POL(c) = 0
POL(f(x_1)) = x_1
POL(found(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   top(mark(x)) -> top(check(x))
   top(active(c)) -> top(mark(c))
   top(found(x)) -> top(active(x))


 This transformation is resulting in one new subcycle:


SCC6.MRR1:

TOP(found(x)) -> TOP(active(x))
TOP(mark(x)) -> TOP(check(x))
TOP(active(c)) -> TOP(mark(c))

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

Additionally, the following rules can be oriented: 

   active(f(x)) -> mark(x)
   active(f(x)) -> f(active(x))
   check(x) -> start(match(f(X), x))
   check(f(x)) -> f(check(x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   start(ok(x)) -> found(x)
   proper(f(x)) -> f(proper(x))
   proper(c) -> ok(c)
   f(found(x)) -> found(f(x))
   f(ok(x)) -> ok(f(x))
   f(mark(x)) -> mark(f(x))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(active(x_1)) = x_1
POL(start(x_1)) = x_1
POL(proper(x_1)) = 1
POL(X) = 1
POL(TOP(x_1)) = 1 + x_1
POL(mark(x_1)) = 0
POL(match(x_1, x_2)) = x_1
POL(check(x_1)) = 0
POL(ok(x_1)) = x_1
POL(f(x_1)) = 0
POL(c) = 1
POL(found(x_1)) = x_1

 resulting in one subcycle.


SCC6.MRR1.Polo1:

TOP(mark(x)) -> TOP(check(x))
TOP(found(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(start(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(X) = 0
POL(TOP(x_1)) = 1 + x_1
POL(mark(x_1)) = 1 + x_1
POL(match(x_1, x_2)) = x_1 + x_2
POL(check(x_1)) = 1 + x_1
POL(ok(x_1)) = x_1
POL(f(x_1)) = 1 + x_1
POL(c) = 0
POL(found(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   match(f(x), f(y)) -> f(match(x, y))


 This transformation is resulting in one new subcycle:


SCC6.MRR1.Polo1.MRR1:

TOP(found(x)) -> TOP(active(x))
TOP(mark(x)) -> TOP(check(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: 

   active(f(x)) -> mark(x)
   active(f(x)) -> f(active(x))
   check(x) -> start(match(f(X), x))
   check(f(x)) -> f(check(x))
   match(X, x) -> proper(x)
   start(ok(x)) -> found(x)
   proper(f(x)) -> f(proper(x))
   proper(c) -> ok(c)
   f(found(x)) -> found(f(x))
   f(ok(x)) -> ok(f(x))
   f(mark(x)) -> mark(f(x))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(active(x_1)) = x_1
POL(start(x_1)) = 1 + x_1
POL(proper(x_1)) = x_1
POL(X) = 0
POL(TOP(x_1)) = 1 + x_1
POL(mark(x_1)) = 1 + x_1
POL(match(x_1, x_2)) = x_2
POL(check(x_1)) = 1 + x_1
POL(ok(x_1)) = x_1
POL(f(x_1)) = 1 + x_1
POL(c) = 0
POL(found(x_1)) = 1 + x_1

 resulting in one subcycle.


SCC6.MRR1.Polo1.MRR1.Polo1:

TOP(mark(x)) -> TOP(check(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(start(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(X) = 0
POL(TOP(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(match(x_1, x_2)) = x_1 + x_2
POL(check(x_1)) = x_1
POL(ok(x_1)) = x_1
POL(f(x_1)) = x_1
POL(c) = 0
POL(found(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   active(f(x)) -> mark(x)
   active(f(x)) -> f(active(x))


 This transformation is resulting in one new subcycle:


SCC6.MRR1.Polo1.MRR1.Polo1.MRR1:

TOP(mark(x)) -> TOP(check(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(start(x_1)) = x_1
POL(proper(x_1)) = x_1
POL(X) = 0
POL(TOP(x_1)) = x_1
POL(mark(x_1)) = 1 + x_1
POL(match(x_1, x_2)) = x_1 + x_2
POL(check(x_1)) = x_1
POL(ok(x_1)) = x_1
POL(f(x_1)) = x_1
POL(c) = 0
POL(found(x_1)) = x_1

The following Dependency Pairs can be deleted:

   TOP(mark(x)) -> TOP(check(x))



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 1.966 seconds.