Term Rewriting System R:

   [Y, X]
   minus(n__0, Y) -> 0
   minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
   geq(X, n__0) -> true
   geq(n__0, n__s(Y)) -> false
   geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
   div(0, n__s(Y)) -> 0
   div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   0 -> n__0
   s(X) -> n__s(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(X)
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ACTIVATE(n__0) -> 0'
   ACTIVATE(n__s(X)) -> S(X)
   DIV(s(X), n__s(Y)) -> IF(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
   DIV(s(X), n__s(Y)) -> GEQ(X, activate(Y))
   DIV(s(X), n__s(Y)) -> ACTIVATE(Y)
   DIV(s(X), n__s(Y)) -> DIV(minus(X, activate(Y)), n__s(activate(Y)))
   DIV(s(X), n__s(Y)) -> MINUS(X, activate(Y))
   IF(false, X, Y) -> ACTIVATE(Y)
   IF(true, X, Y) -> ACTIVATE(X)
   GEQ(n__s(X), n__s(Y)) -> GEQ(activate(X), activate(Y))
   GEQ(n__s(X), n__s(Y)) -> ACTIVATE(X)
   GEQ(n__s(X), n__s(Y)) -> ACTIVATE(Y)
   MINUS(n__s(X), n__s(Y)) -> MINUS(activate(X), activate(Y))
   MINUS(n__s(X), n__s(Y)) -> ACTIVATE(X)
   MINUS(n__s(X), n__s(Y)) -> ACTIVATE(Y)
   MINUS(n__0, Y) -> 0'

Furthermore, R contains three SCCs.


SCC1:

GEQ(n__s(X), n__s(Y)) -> GEQ(activate(X), activate(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(activate(x_1)) = x_1
POL(s(x_1)) = x_1
POL(n__0) = 0
POL(GEQ(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(n__s(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   div(0, n__s(Y)) -> 0
   div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
   if(false, X, Y) -> activate(Y)
   if(true, X, Y) -> activate(X)
   geq(X, n__0) -> true
   geq(n__0, n__s(Y)) -> false
   geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
   minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
   minus(n__0, Y) -> 0


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

GEQ(n__s(X), n__s(Y)) -> GEQ(activate(X), activate(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(activate(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(n__0) = 0
POL(GEQ(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(n__s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   GEQ(n__s(X), n__s(Y)) -> GEQ(activate(X), activate(Y))



 This transformation is resulting in no new subcycles.


SCC2:

MINUS(n__s(X), n__s(Y)) -> MINUS(activate(X), activate(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(activate(x_1)) = x_1
POL(s(x_1)) = x_1
POL(n__0) = 0
POL(MINUS(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(n__s(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   div(0, n__s(Y)) -> 0
   div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0)
   if(false, X, Y) -> activate(Y)
   if(true, X, Y) -> activate(X)
   geq(X, n__0) -> true
   geq(n__0, n__s(Y)) -> false
   geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
   minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
   minus(n__0, Y) -> 0


 This transformation is resulting in one new subcycle:


SCC2.MRR1:

MINUS(n__s(X), n__s(Y)) -> MINUS(activate(X), activate(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(activate(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(n__0) = 0
POL(MINUS(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0
POL(n__s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   MINUS(n__s(X), n__s(Y)) -> MINUS(activate(X), activate(Y))



 This transformation is resulting in no new subcycles.


SCC3:

DIV(s(X), n__s(Y)) -> DIV(minus(X, activate(Y)), n__s(activate(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(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
   minus(n__0, Y) -> 0
   0 -> n__0


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(activate(x_1)) = 0
POL(s(x_1)) = 1
POL(DIV(x_1, x_2)) = x_1
POL(n__0) = 0
POL(minus(x_1, x_2)) = 0
POL(0) = 0
POL(n__s(x_1)) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 1.37 seconds.