Term Rewriting System R:

   [X, Y, Z, X1, X2]
   a__first(0, X) -> nil
   a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))
   a__first(X1, X2) -> first(X1, X2)
   a__from(X) -> cons(mark(X), from(s(X)))
   a__from(X) -> from(X)
   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(0) -> 0
   mark(nil) -> nil
   mark(s(X)) -> s(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)

Termination of R to be shown.


R contains the following Dependency Pairs: 


   A__FIRST(s(X), cons(Y, Z)) -> MARK(Y)
   A__FROM(X) -> MARK(X)
   MARK(s(X)) -> MARK(X)
   MARK(first(X1, X2)) -> A__FIRST(mark(X1), mark(X2))
   MARK(first(X1, X2)) -> MARK(X1)
   MARK(first(X1, X2)) -> MARK(X2)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)

Furthermore, R contains one SCC.


SCC1:

MARK(from(X)) -> MARK(X)
A__FROM(X) -> MARK(X)
MARK(from(X)) -> A__FROM(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> A__FIRST(mark(X1), mark(X2))
MARK(s(X)) -> MARK(X)
A__FIRST(s(X), cons(Y, Z)) -> MARK(Y)

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(first(X1, X2)) -> a__first(mark(X1), mark(X2))
   mark(nil) -> nil
   mark(0) -> 0
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(from(X)) -> a__from(mark(X))
   a__first(0, X) -> nil
   a__first(X1, X2) -> first(X1, X2)
   a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(first(x_1, x_2)) = x_1 + x_2
POL(nil) = 0
POL(s(x_1)) = x_1
POL(A__FROM(x_1)) = x_1
POL(A__FIRST(x_1, x_2)) = x_2
POL(a__from(x_1)) = 1 + x_1
POL(mark(x_1)) = x_1
POL(MARK(x_1)) = x_1
POL(from(x_1)) = 1 + x_1
POL(a__first(x_1, x_2)) = x_1 + x_2
POL(0) = 0
POL(cons(x_1, x_2)) = x_1

 resulting in one subcycle.


SCC1.Polo1:

MARK(cons(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
A__FIRST(s(X), cons(Y, Z)) -> MARK(Y)
MARK(first(X1, X2)) -> A__FIRST(mark(X1), mark(X2))
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(first(X1, X2)) -> a__first(mark(X1), mark(X2))
   mark(nil) -> nil
   mark(0) -> 0
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(from(X)) -> a__from(mark(X))
   a__first(0, X) -> nil
   a__first(X1, X2) -> first(X1, X2)
   a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(first(x_1, x_2)) = 1 + x_1 + x_2
POL(nil) = 0
POL(s(x_1)) = 1 + x_1
POL(A__FIRST(x_1, x_2)) = x_2
POL(a__from(x_1)) = 1 + x_1
POL(mark(x_1)) = x_1
POL(MARK(x_1)) = x_1
POL(from(x_1)) = 1 + x_1
POL(a__first(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(cons(x_1, x_2)) = 1 + x_1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 1.285 seconds.