Term Rewriting System R:

   [X, Y, X1, X2]
   a__from(X) -> cons(mark(X), from(s(X)))
   a__from(X) -> from(X)
   a__length(nil) -> 0
   a__length(cons(X, Y)) -> s(a__length1(Y))
   a__length(X) -> length(X)
   a__length1(X) -> a__length(X)
   a__length1(X) -> length1(X)
   mark(from(X)) -> a__from(mark(X))
   mark(length(X)) -> a__length(X)
   mark(length1(X)) -> a__length1(X)
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(0) -> 0

Termination of R to be shown.


R contains the following Dependency Pairs: 


   MARK(s(X)) -> MARK(X)
   MARK(length(X)) -> A__LENGTH(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)
   MARK(length1(X)) -> A__LENGTH1(X)
   A__LENGTH(cons(X, Y)) -> A__LENGTH1(Y)
   A__LENGTH1(X) -> A__LENGTH(X)
   A__FROM(X) -> MARK(X)

Furthermore, R contains two SCCs.


SCC1:

A__LENGTH1(X) -> A__LENGTH(X)
A__LENGTH(cons(X, Y)) -> A__LENGTH1(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(A__LENGTH1(x_1)) = x_1
POL(A__LENGTH(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1 + x_2

The following Dependency Pairs can be deleted:

   A__LENGTH(cons(X, Y)) -> A__LENGTH1(Y)



 This transformation is resulting in no new subcycles.


SCC2:

MARK(from(X)) -> MARK(X)
A__FROM(X) -> MARK(X)
MARK(from(X)) -> A__FROM(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(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: 

   a__from(X) -> cons(mark(X), from(s(X)))
   a__from(X) -> from(X)
   mark(s(X)) -> s(mark(X))
   mark(length(X)) -> a__length(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(from(X)) -> a__from(mark(X))
   mark(0) -> 0
   mark(length1(X)) -> a__length1(X)
   a__length1(X) -> length1(X)
   a__length1(X) -> a__length(X)
   a__length(X) -> length(X)
   a__length(nil) -> 0
   a__length(cons(X, Y)) -> s(a__length1(Y))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(s(x_1)) = x_1
POL(length(x_1)) = 0
POL(A__FROM(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(MARK(x_1)) = x_1
POL(a__length(x_1)) = 0
POL(0) = 0
POL(cons(x_1, x_2)) = x_1
POL(length1(x_1)) = 0
POL(a__length1(x_1)) = 0
POL(a__from(x_1)) = 1 + x_1
POL(from(x_1)) = 1 + x_1

 resulting in one subcycle.


SCC2.Polo1:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(s(X)) -> MARK(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(s(x_1)) = 1 + x_1
POL(MARK(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(s(X)) -> MARK(X)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.863 seconds.