Term Rewriting System R:

   [Z, X, Y, X1, X2]
   a__fst(0, Z) -> nil
   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
   a__fst(X1, X2) -> fst(X1, X2)
   a__from(X) -> cons(mark(X), from(s(X)))
   a__from(X) -> from(X)
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   a__add(X1, X2) -> add(X1, X2)
   a__len(nil) -> 0
   a__len(cons(X, Z)) -> s(len(Z))
   a__len(X) -> len(X)
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(len(X)) -> a__len(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)

Termination of R to be shown.


R contains the following Dependency Pairs: 


   MARK(len(X)) -> A__LEN(mark(X))
   MARK(len(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(from(X)) -> A__FROM(mark(X))
   MARK(from(X)) -> MARK(X)
   MARK(add(X1, X2)) -> A__ADD(mark(X1), mark(X2))
   MARK(add(X1, X2)) -> MARK(X1)
   MARK(add(X1, X2)) -> MARK(X2)
   MARK(fst(X1, X2)) -> A__FST(mark(X1), mark(X2))
   MARK(fst(X1, X2)) -> MARK(X1)
   MARK(fst(X1, X2)) -> MARK(X2)
   A__FST(s(X), cons(Y, Z)) -> MARK(Y)
   A__FROM(X) -> MARK(X)
   A__ADD(0, X) -> MARK(X)

Furthermore, R contains one SCC.


SCC1:

MARK(fst(X1, X2)) -> MARK(X2)
MARK(fst(X1, X2)) -> MARK(X1)
A__FST(s(X), cons(Y, Z)) -> MARK(Y)
MARK(fst(X1, X2)) -> A__FST(mark(X1), mark(X2))
MARK(add(X1, X2)) -> MARK(X2)
MARK(add(X1, X2)) -> MARK(X1)
A__ADD(0, X) -> MARK(X)
MARK(add(X1, X2)) -> A__ADD(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
A__FROM(X) -> MARK(X)
MARK(from(X)) -> A__FROM(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(len(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)
   a__fst(X1, X2) -> fst(X1, X2)
   a__fst(0, Z) -> nil
   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
   a__add(X1, X2) -> add(X1, X2)
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   mark(len(X)) -> a__len(mark(X))
   mark(nil) -> nil
   mark(s(X)) -> s(X)
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(from(X)) -> a__from(mark(X))
   mark(0) -> 0
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   a__len(X) -> len(X)
   a__len(nil) -> 0
   a__len(cons(X, Z)) -> s(len(Z))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(add(x_1, x_2)) = x_1 + x_2
POL(A__FROM(x_1)) = x_1
POL(s(x_1)) = 0
POL(A__FST(x_1, x_2)) = x_2
POL(MARK(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(a__add(x_1, x_2)) = x_1 + x_2
POL(fst(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(cons(x_1, x_2)) = x_1
POL(A__ADD(x_1, x_2)) = x_2
POL(a__from(x_1)) = x_1
POL(a__fst(x_1, x_2)) = 1 + x_1 + x_2
POL(from(x_1)) = x_1
POL(len(x_1)) = x_1
POL(a__len(x_1)) = x_1

 resulting in one subcycle.


SCC1.Polo1:

MARK(add(X1, X2)) -> MARK(X2)
MARK(add(X1, X2)) -> MARK(X1)
A__ADD(0, X) -> MARK(X)
MARK(add(X1, X2)) -> A__ADD(mark(X1), mark(X2))
MARK(from(X)) -> MARK(X)
A__FROM(X) -> MARK(X)
MARK(from(X)) -> A__FROM(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(len(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)
   a__fst(X1, X2) -> fst(X1, X2)
   a__fst(0, Z) -> nil
   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
   a__add(X1, X2) -> add(X1, X2)
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   mark(len(X)) -> a__len(mark(X))
   mark(nil) -> nil
   mark(s(X)) -> s(X)
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(from(X)) -> a__from(mark(X))
   mark(0) -> 0
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   a__len(X) -> len(X)
   a__len(nil) -> 0
   a__len(cons(X, Z)) -> s(len(Z))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(A__FROM(x_1)) = x_1
POL(s(x_1)) = 0
POL(MARK(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(a__add(x_1, x_2)) = 1 + x_1 + x_2
POL(fst(x_1, x_2)) = x_2
POL(0) = 0
POL(cons(x_1, x_2)) = x_1
POL(A__ADD(x_1, x_2)) = x_2
POL(a__from(x_1)) = x_1
POL(a__fst(x_1, x_2)) = x_2
POL(from(x_1)) = x_1
POL(len(x_1)) = x_1
POL(a__len(x_1)) = x_1

 resulting in one subcycle.


SCC1.Polo1.Polo1:

MARK(from(X)) -> MARK(X)
A__FROM(X) -> MARK(X)
MARK(from(X)) -> A__FROM(mark(X))
MARK(cons(X1, X2)) -> MARK(X1)
MARK(len(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)
   a__fst(X1, X2) -> fst(X1, X2)
   a__fst(0, Z) -> nil
   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
   a__add(X1, X2) -> add(X1, X2)
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   mark(len(X)) -> a__len(mark(X))
   mark(nil) -> nil
   mark(s(X)) -> s(X)
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(from(X)) -> a__from(mark(X))
   mark(0) -> 0
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   a__len(X) -> len(X)
   a__len(nil) -> 0
   a__len(cons(X, Z)) -> s(len(Z))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(add(x_1, x_2)) = x_2
POL(A__FROM(x_1)) = x_1
POL(s(x_1)) = 0
POL(MARK(x_1)) = x_1
POL(mark(x_1)) = x_1
POL(a__add(x_1, x_2)) = x_2
POL(fst(x_1, x_2)) = x_2
POL(0) = 0
POL(cons(x_1, x_2)) = x_1
POL(a__from(x_1)) = 1 + x_1
POL(a__fst(x_1, x_2)) = x_2
POL(from(x_1)) = 1 + x_1
POL(len(x_1)) = x_1
POL(a__len(x_1)) = x_1

 resulting in one subcycle.


SCC1.Polo1.Polo1.Polo1:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 2.664 seconds.