Term Rewriting System R:

   [N, X, Y, X1, X2, Z]
   a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N)))
   a__terms(X) -> terms(X)
   a__sqr(0) -> 0
   a__sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   a__sqr(X) -> sqr(X)
   a__dbl(0) -> 0
   a__dbl(s(X)) -> s(s(dbl(X)))
   a__dbl(X) -> dbl(X)
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   a__add(X1, X2) -> add(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)
   mark(terms(X)) -> a__terms(mark(X))
   mark(sqr(X)) -> a__sqr(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(dbl(X)) -> a__dbl(mark(X))
   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(recip(X)) -> recip(mark(X))
   mark(s(X)) -> s(X)
   mark(0) -> 0
   mark(nil) -> nil

Termination of R to be shown.


R contains the following Dependency Pairs: 


   A__FIRST(s(X), cons(Y, Z)) -> MARK(Y)
   MARK(first(X1, X2)) -> A__FIRST(mark(X1), mark(X2))
   MARK(first(X1, X2)) -> MARK(X1)
   MARK(first(X1, X2)) -> MARK(X2)
   MARK(dbl(X)) -> A__DBL(mark(X))
   MARK(dbl(X)) -> MARK(X)
   MARK(terms(X)) -> A__TERMS(mark(X))
   MARK(terms(X)) -> MARK(X)
   MARK(sqr(X)) -> A__SQR(mark(X))
   MARK(sqr(X)) -> MARK(X)
   MARK(cons(X1, X2)) -> MARK(X1)
   MARK(recip(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)
   A__TERMS(N) -> A__SQR(mark(N))
   A__TERMS(N) -> MARK(N)
   A__ADD(0, X) -> MARK(X)

Furthermore, R contains one SCC.


SCC1:

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(recip(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sqr(X)) -> MARK(X)
MARK(terms(X)) -> MARK(X)
A__TERMS(N) -> MARK(N)
MARK(terms(X)) -> A__TERMS(mark(X))
MARK(dbl(X)) -> MARK(X)
MARK(first(X1, X2)) -> MARK(X2)
MARK(first(X1, X2)) -> MARK(X1)
MARK(first(X1, X2)) -> A__FIRST(mark(X1), mark(X2))
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__terms(X) -> terms(X)
   a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N)))
   a__add(X1, X2) -> add(X1, X2)
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
   mark(nil) -> nil
   mark(s(X)) -> s(X)
   mark(dbl(X)) -> a__dbl(mark(X))
   mark(terms(X)) -> a__terms(mark(X))
   mark(sqr(X)) -> a__sqr(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(recip(X)) -> recip(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(0) -> 0
   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__dbl(X) -> dbl(X)
   a__dbl(s(X)) -> s(s(dbl(X)))
   a__dbl(0) -> 0
   a__sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   a__sqr(0) -> 0
   a__sqr(X) -> sqr(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(A__TERMS(x_1)) = x_1
POL(add(x_1, x_2)) = 1 + x_1 + x_2
POL(nil) = 0
POL(s(x_1)) = 0
POL(dbl(x_1)) = x_1
POL(MARK(x_1)) = x_1
POL(a__add(x_1, x_2)) = 1 + x_1 + x_2
POL(mark(x_1)) = x_1
POL(a__first(x_1, x_2)) = x_1 + x_2
POL(0) = 0
POL(a__dbl(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1
POL(first(x_1, x_2)) = x_1 + x_2
POL(A__FIRST(x_1, x_2)) = x_2
POL(a__terms(x_1)) = x_1
POL(terms(x_1)) = x_1
POL(A__ADD(x_1, x_2)) = x_2
POL(sqr(x_1)) = x_1
POL(recip(x_1)) = x_1
POL(a__sqr(x_1)) = x_1

 resulting in one subcycle.


SCC1.Polo1:

MARK(recip(X)) -> MARK(X)
MARK(cons(X1, X2)) -> MARK(X1)
MARK(sqr(X)) -> MARK(X)
MARK(terms(X)) -> MARK(X)
A__TERMS(N) -> MARK(N)
MARK(terms(X)) -> A__TERMS(mark(X))
MARK(dbl(X)) -> MARK(X)
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))

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__terms(X) -> terms(X)
   a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N)))
   a__add(X1, X2) -> add(X1, X2)
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
   mark(nil) -> nil
   mark(s(X)) -> s(X)
   mark(dbl(X)) -> a__dbl(mark(X))
   mark(terms(X)) -> a__terms(mark(X))
   mark(sqr(X)) -> a__sqr(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(recip(X)) -> recip(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(0) -> 0
   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__dbl(X) -> dbl(X)
   a__dbl(s(X)) -> s(s(dbl(X)))
   a__dbl(0) -> 0
   a__sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   a__sqr(0) -> 0
   a__sqr(X) -> sqr(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(A__TERMS(x_1)) = x_1
POL(add(x_1, x_2)) = x_2
POL(nil) = 0
POL(s(x_1)) = 0
POL(dbl(x_1)) = x_1
POL(MARK(x_1)) = x_1
POL(a__add(x_1, x_2)) = x_2
POL(mark(x_1)) = x_1
POL(a__first(x_1, x_2)) = x_1 + x_2
POL(0) = 0
POL(a__dbl(x_1)) = x_1
POL(cons(x_1, x_2)) = x_1
POL(first(x_1, x_2)) = x_1 + x_2
POL(A__FIRST(x_1, x_2)) = x_2
POL(a__terms(x_1)) = 1 + x_1
POL(terms(x_1)) = 1 + x_1
POL(sqr(x_1)) = x_1
POL(recip(x_1)) = 1 + x_1
POL(a__sqr(x_1)) = x_1

 resulting in one subcycle.


SCC1.Polo1.Polo1:

MARK(sqr(X)) -> MARK(X)
MARK(dbl(X)) -> MARK(X)
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(cons(X1, X2)) -> MARK(X1)

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__terms(X) -> terms(X)
   a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N)))
   a__add(X1, X2) -> add(X1, X2)
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   mark(first(X1, X2)) -> a__first(mark(X1), mark(X2))
   mark(nil) -> nil
   mark(s(X)) -> s(X)
   mark(dbl(X)) -> a__dbl(mark(X))
   mark(terms(X)) -> a__terms(mark(X))
   mark(sqr(X)) -> a__sqr(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(recip(X)) -> recip(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(0) -> 0
   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__dbl(X) -> dbl(X)
   a__dbl(s(X)) -> s(s(dbl(X)))
   a__dbl(0) -> 0
   a__sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   a__sqr(0) -> 0
   a__sqr(X) -> sqr(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(add(x_1, x_2)) = x_2
POL(nil) = 0
POL(s(x_1)) = 0
POL(dbl(x_1)) = 1 + x_1
POL(MARK(x_1)) = x_1
POL(a__add(x_1, x_2)) = x_2
POL(mark(x_1)) = x_1
POL(a__first(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(a__dbl(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = 1 + x_1
POL(first(x_1, x_2)) = 1 + x_1 + x_2
POL(A__FIRST(x_1, x_2)) = x_2
POL(a__terms(x_1)) = 1
POL(terms(x_1)) = 1
POL(sqr(x_1)) = 1 + x_1
POL(recip(x_1)) = 0
POL(a__sqr(x_1)) = 1 + x_1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 3.620 seconds.