Term Rewriting System R:

   [X, XS, N, Y, YS, X1, X2]
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
   zWquot(X1, X2) -> n__zWquot(X1, X2)
   s(X) -> n__s(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(n__zWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ACTIVATE(n__zWquot(X1, X2)) -> ZWQUOT(activate(X1), activate(X2))
   ACTIVATE(n__zWquot(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__zWquot(X1, X2)) -> ACTIVATE(X2)
   ACTIVATE(n__s(X)) -> S(activate(X))
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__from(X)) -> FROM(activate(X))
   ACTIVATE(n__from(X)) -> ACTIVATE(X)
   ZWQUOT(cons(X, XS), cons(Y, YS)) -> QUOT(X, Y)
   ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
   ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
   QUOT(s(X), s(Y)) -> S(quot(minus(X, Y), s(Y)))
   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
   QUOT(s(X), s(Y)) -> MINUS(X, Y)
   MINUS(s(X), s(Y)) -> MINUS(X, Y)
   SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
   SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)

Furthermore, R contains four SCCs.


SCC1:

MINUS(s(X), s(Y)) -> MINUS(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(s(x_1)) = 1 + x_1
POL(MINUS(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   MINUS(s(X), s(Y)) -> MINUS(X, Y)



 This transformation is resulting in no new subcycles.


SCC2:

QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(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: 

   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1
POL(minus(x_1, x_2)) = 0
POL(QUOT(x_1, x_2)) = x_1
POL(0) = 0
POL(n__s(x_1)) = 0

 resulting in no subcycles.


SCC3:

ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(n__from(X)) -> ACTIVATE(X)
ACTIVATE(n__s(X)) -> ACTIVATE(X)
ACTIVATE(n__zWquot(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n__zWquot(X1, X2)) -> ACTIVATE(X1)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(n__zWquot(X1, X2)) -> ZWQUOT(activate(X1), activate(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: 

   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
   zWquot(XS, nil) -> nil
   zWquot(X1, X2) -> n__zWquot(X1, X2)
   zWquot(nil, XS) -> nil
   activate(n__zWquot(X1, X2)) -> zWquot(activate(X1), activate(X2))
   activate(X) -> X
   activate(n__s(X)) -> s(activate(X))
   activate(n__from(X)) -> from(activate(X))
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   s(X) -> n__s(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(activate(x_1)) = x_1
POL(s(x_1)) = x_1
POL(n__zWquot(x_1, x_2)) = 1 + x_1 + x_2
POL(minus(x_1, x_2)) = 0
POL(ZWQUOT(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(cons(x_1, x_2)) = x_2
POL(quot(x_1, x_2)) = 0
POL(zWquot(x_1, x_2)) = 1 + x_1 + x_2
POL(n__from(x_1)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(from(x_1)) = x_1
POL(n__s(x_1)) = x_1

 resulting in one subcycle.


SCC3.Polo1:

ACTIVATE(n__s(X)) -> ACTIVATE(X)
ACTIVATE(n__from(X)) -> ACTIVATE(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(n__from(x_1)) = 1 + x_1
POL(ACTIVATE(x_1)) = 1 + x_1
POL(n__s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__from(X)) -> ACTIVATE(X)



 This transformation is resulting in no new subcycles.


SCC4:

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(activate(x_1)) = 0
POL(s(x_1)) = 1 + x_1
POL(n__zWquot(x_1, x_2)) = 0
POL(minus(x_1, x_2)) = 0
POL(0) = 0
POL(cons(x_1, x_2)) = 0
POL(quot(x_1, x_2)) = 0
POL(zWquot(x_1, x_2)) = 0
POL(n__from(x_1)) = 0
POL(SEL(x_1, x_2)) = x_1
POL(from(x_1)) = 0
POL(n__s(x_1)) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 1.385 seconds.