Term Rewriting System R:
[Y, X]
minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

MINUS(n0, Y) -> 0'
MINUS(ns(X), ns(Y)) -> MINUS(activate(X), activate(Y))
MINUS(ns(X), ns(Y)) -> ACTIVATE(X)
MINUS(ns(X), ns(Y)) -> ACTIVATE(Y)
GEQ(ns(X), ns(Y)) -> GEQ(activate(X), activate(Y))
GEQ(ns(X), ns(Y)) -> ACTIVATE(X)
GEQ(ns(X), ns(Y)) -> ACTIVATE(Y)
DIV(s(X), ns(Y)) -> IF(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
DIV(s(X), ns(Y)) -> GEQ(X, activate(Y))
DIV(s(X), ns(Y)) -> ACTIVATE(Y)
DIV(s(X), ns(Y)) -> DIV(minus(X, activate(Y)), ns(activate(Y)))
DIV(s(X), ns(Y)) -> MINUS(X, activate(Y))
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(n0) -> 0'
ACTIVATE(ns(X)) -> S(X)

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pair:

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


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(X), ns(Y)) -> MINUS(activate(X), activate(Y))
six new Dependency Pairs are created:

MINUS(ns(n0), ns(Y)) -> MINUS(0, activate(Y))
MINUS(ns(ns(X'')), ns(Y)) -> MINUS(s(X''), activate(Y))
MINUS(ns(X''), ns(Y)) -> MINUS(X'', activate(Y))
MINUS(ns(X), ns(n0)) -> MINUS(activate(X), 0)
MINUS(ns(X), ns(ns(X''))) -> MINUS(activate(X), s(X''))
MINUS(ns(X), ns(Y')) -> MINUS(activate(X), Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X), ns(Y')) -> MINUS(activate(X), Y')
MINUS(ns(X), ns(ns(X''))) -> MINUS(activate(X), s(X''))
MINUS(ns(X), ns(n0)) -> MINUS(activate(X), 0)
MINUS(ns(X''), ns(Y)) -> MINUS(X'', activate(Y))
MINUS(ns(ns(X'')), ns(Y)) -> MINUS(s(X''), activate(Y))
MINUS(ns(n0), ns(Y)) -> MINUS(0, activate(Y))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(n0), ns(Y)) -> MINUS(0, activate(Y))
four new Dependency Pairs are created:

MINUS(ns(n0), ns(Y)) -> MINUS(n0, activate(Y))
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(n0), ns(ns(X'))) -> MINUS(0, s(X'))
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 5
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(n0), ns(ns(X'))) -> MINUS(0, s(X'))
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X), ns(ns(X''))) -> MINUS(activate(X), s(X''))
MINUS(ns(X), ns(n0)) -> MINUS(activate(X), 0)
MINUS(ns(X''), ns(Y)) -> MINUS(X'', activate(Y))
MINUS(ns(ns(X'')), ns(Y)) -> MINUS(s(X''), activate(Y))
MINUS(ns(X), ns(Y')) -> MINUS(activate(X), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X'')), ns(Y)) -> MINUS(s(X''), activate(Y))
four new Dependency Pairs are created:

MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 6
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
MINUS(ns(n0), ns(ns(X'))) -> MINUS(0, s(X'))
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X), ns(Y')) -> MINUS(activate(X), Y')
MINUS(ns(X), ns(ns(X''))) -> MINUS(activate(X), s(X''))
MINUS(ns(X), ns(n0)) -> MINUS(activate(X), 0)
MINUS(ns(X''), ns(Y)) -> MINUS(X'', activate(Y))
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(X''), ns(Y)) -> MINUS(X'', activate(Y))
three new Dependency Pairs are created:

MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 7
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(n0), ns(ns(X'))) -> MINUS(0, s(X'))
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X), ns(Y')) -> MINUS(activate(X), Y')
MINUS(ns(X), ns(ns(X''))) -> MINUS(activate(X), s(X''))
MINUS(ns(X), ns(n0)) -> MINUS(activate(X), 0)
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(X), ns(n0)) -> MINUS(activate(X), 0)
four new Dependency Pairs are created:

MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(X), ns(n0)) -> MINUS(activate(X), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 8
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(n0), ns(ns(X'))) -> MINUS(0, s(X'))
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X), ns(Y')) -> MINUS(activate(X), Y')
MINUS(ns(X), ns(ns(X''))) -> MINUS(activate(X), s(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(X), ns(ns(X''))) -> MINUS(activate(X), s(X''))
four new Dependency Pairs are created:

MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 9
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(n0), ns(ns(X'))) -> MINUS(0, s(X'))
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X), ns(Y')) -> MINUS(activate(X), Y')
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(X), ns(Y')) -> MINUS(activate(X), Y')
three new Dependency Pairs are created:

MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 10
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(n0), ns(ns(X'))) -> MINUS(0, s(X'))
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
two new Dependency Pairs are created:

MINUS(ns(n0), ns(n0)) -> MINUS(n0, 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 11
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(n0), ns(ns(X'))) -> MINUS(0, s(X'))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(n0), ns(ns(X'))) -> MINUS(0, s(X'))
two new Dependency Pairs are created:

MINUS(ns(n0), ns(ns(X'))) -> MINUS(n0, s(X'))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 12
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
one new Dependency Pair is created:

MINUS(ns(n0), ns(Y')) -> MINUS(n0, Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 13
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
three new Dependency Pairs are created:

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 14
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
two new Dependency Pairs are created:

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 15
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
two new Dependency Pairs are created:

MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 16
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
one new Dependency Pair is created:

MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 17
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
one new Dependency Pair is created:

MINUS(ns(X''), ns(n0)) -> MINUS(X'', n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 18
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(X''), ns(ns(X'))) -> MINUS(X'', s(X'))
one new Dependency Pair is created:

MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 19
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(n0), ns(n0)) -> MINUS(0, 0)
two new Dependency Pairs are created:

MINUS(ns(n0), ns(n0)) -> MINUS(n0, 0)
MINUS(ns(n0), ns(n0)) -> MINUS(0, n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 20
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), 0)
two new Dependency Pairs are created:

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X'')), ns(n0)) -> MINUS(s(X''), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 21
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(X''), ns(n0)) -> MINUS(X'', 0)
one new Dependency Pair is created:

MINUS(ns(X''), ns(n0)) -> MINUS(X'', n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 22
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, s(X''))
two new Dependency Pairs are created:

MINUS(ns(n0), ns(ns(X''))) -> MINUS(n0, s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 23
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(s(X'''), s(X''))
two new Dependency Pairs are created:

MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 24
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(X0), ns(ns(X''))) -> MINUS(X0, s(X''))
one new Dependency Pair is created:

MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 25
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
three new Dependency Pairs are created:

MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 26
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(n0), ns(Y')) -> MINUS(0, Y')
one new Dependency Pair is created:

MINUS(ns(n0), ns(Y')) -> MINUS(n0, Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 27
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X'')), ns(Y')) -> MINUS(s(X''), Y')
one new Dependency Pair is created:

MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 28
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(n0), ns(ns(X''))) -> MINUS(0, ns(X''))
one new Dependency Pair is created:

MINUS(ns(n0), ns(ns(X''))) -> MINUS(n0, ns(X''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 29
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
one new Dependency Pair is created:

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 30
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
one new Dependency Pair is created:

MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 31
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
one new Dependency Pair is created:

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 32
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X''')), ns(ns(X'))) -> MINUS(ns(X'''), s(X'))
one new Dependency Pair is created:

MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 33
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
one new Dependency Pair is created:

MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 34
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), 0)
one new Dependency Pair is created:

MINUS(ns(ns(X''')), ns(n0)) -> MINUS(ns(X'''), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 35
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
one new Dependency Pair is created:

MINUS(ns(n0), ns(ns(X'''))) -> MINUS(n0, ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 36
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X'''')), ns(ns(X''))) -> MINUS(ns(X''''), s(X''))
one new Dependency Pair is created:

MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 37
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X''')), ns(ns(X''''))) -> MINUS(s(X'''), ns(X''''))
one new Dependency Pair is created:

MINUS(ns(ns(X''''')), ns(ns(X''''))) -> MINUS(ns(X'''''), ns(X''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 38
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X''''')), ns(ns(X''''))) -> MINUS(ns(X'''''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(n0), ns(ns(X'''))) -> MINUS(0, ns(X'''))
one new Dependency Pair is created:

MINUS(ns(n0), ns(ns(X'''))) -> MINUS(n0, ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 39
                 ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))
MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X''''')), ns(ns(X''''))) -> MINUS(ns(X'''''), ns(X''''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

MINUS(ns(ns(X'')), ns(ns(X'''))) -> MINUS(s(X''), ns(X'''))
one new Dependency Pair is created:

MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 40
                 ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))
MINUS(ns(ns(X''''')), ns(ns(X''''))) -> MINUS(ns(X'''''), ns(X''''))
MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')
MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





The following dependency pairs can be strictly oriented:

MINUS(ns(ns(X'''')), ns(ns(X'''))) -> MINUS(ns(X''''), ns(X'''))
MINUS(ns(ns(X''''')), ns(ns(X''''))) -> MINUS(ns(X'''''), ns(X''''))
MINUS(ns(ns(X''')), ns(ns(X''))) -> MINUS(ns(X'''), ns(X''))
MINUS(ns(ns(X''')), ns(Y')) -> MINUS(ns(X'''), Y')
MINUS(ns(X''), ns(ns(X'''))) -> MINUS(X'', ns(X'''))
MINUS(ns(X0), ns(ns(X'''))) -> MINUS(X0, ns(X'''))
MINUS(ns(X''), ns(Y')) -> MINUS(X'', Y')


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
MINUS(x1, x2) -> MINUS(x1, x2)
ns(x1) -> ns(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 4
             ↳Nar
             ...
               →DP Problem 41
                 ↳Dependency Graph
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining


Dependency Pair:


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pair:

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


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(X), ns(Y)) -> GEQ(activate(X), activate(Y))
six new Dependency Pairs are created:

GEQ(ns(n0), ns(Y)) -> GEQ(0, activate(Y))
GEQ(ns(ns(X'')), ns(Y)) -> GEQ(s(X''), activate(Y))
GEQ(ns(X''), ns(Y)) -> GEQ(X'', activate(Y))
GEQ(ns(X), ns(n0)) -> GEQ(activate(X), 0)
GEQ(ns(X), ns(ns(X''))) -> GEQ(activate(X), s(X''))
GEQ(ns(X), ns(Y')) -> GEQ(activate(X), Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X), ns(Y')) -> GEQ(activate(X), Y')
GEQ(ns(X), ns(ns(X''))) -> GEQ(activate(X), s(X''))
GEQ(ns(X), ns(n0)) -> GEQ(activate(X), 0)
GEQ(ns(X''), ns(Y)) -> GEQ(X'', activate(Y))
GEQ(ns(ns(X'')), ns(Y)) -> GEQ(s(X''), activate(Y))
GEQ(ns(n0), ns(Y)) -> GEQ(0, activate(Y))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(n0), ns(Y)) -> GEQ(0, activate(Y))
four new Dependency Pairs are created:

GEQ(ns(n0), ns(Y)) -> GEQ(n0, activate(Y))
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(n0), ns(ns(X'))) -> GEQ(0, s(X'))
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 43
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(n0), ns(ns(X'))) -> GEQ(0, s(X'))
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X), ns(ns(X''))) -> GEQ(activate(X), s(X''))
GEQ(ns(X), ns(n0)) -> GEQ(activate(X), 0)
GEQ(ns(X''), ns(Y)) -> GEQ(X'', activate(Y))
GEQ(ns(ns(X'')), ns(Y)) -> GEQ(s(X''), activate(Y))
GEQ(ns(X), ns(Y')) -> GEQ(activate(X), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X'')), ns(Y)) -> GEQ(s(X''), activate(Y))
four new Dependency Pairs are created:

GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 44
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
GEQ(ns(n0), ns(ns(X'))) -> GEQ(0, s(X'))
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X), ns(Y')) -> GEQ(activate(X), Y')
GEQ(ns(X), ns(ns(X''))) -> GEQ(activate(X), s(X''))
GEQ(ns(X), ns(n0)) -> GEQ(activate(X), 0)
GEQ(ns(X''), ns(Y)) -> GEQ(X'', activate(Y))
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(X''), ns(Y)) -> GEQ(X'', activate(Y))
three new Dependency Pairs are created:

GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 45
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(n0), ns(ns(X'))) -> GEQ(0, s(X'))
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X), ns(Y')) -> GEQ(activate(X), Y')
GEQ(ns(X), ns(ns(X''))) -> GEQ(activate(X), s(X''))
GEQ(ns(X), ns(n0)) -> GEQ(activate(X), 0)
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(X), ns(n0)) -> GEQ(activate(X), 0)
four new Dependency Pairs are created:

GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(X), ns(n0)) -> GEQ(activate(X), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 46
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(n0), ns(ns(X'))) -> GEQ(0, s(X'))
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X), ns(Y')) -> GEQ(activate(X), Y')
GEQ(ns(X), ns(ns(X''))) -> GEQ(activate(X), s(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(X), ns(ns(X''))) -> GEQ(activate(X), s(X''))
four new Dependency Pairs are created:

GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 47
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(n0), ns(ns(X'))) -> GEQ(0, s(X'))
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X), ns(Y')) -> GEQ(activate(X), Y')
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(X), ns(Y')) -> GEQ(activate(X), Y')
three new Dependency Pairs are created:

GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 48
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(n0), ns(ns(X'))) -> GEQ(0, s(X'))
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
two new Dependency Pairs are created:

GEQ(ns(n0), ns(n0)) -> GEQ(n0, 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 49
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(n0), ns(ns(X'))) -> GEQ(0, s(X'))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(n0), ns(ns(X'))) -> GEQ(0, s(X'))
two new Dependency Pairs are created:

GEQ(ns(n0), ns(ns(X'))) -> GEQ(n0, s(X'))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 50
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
one new Dependency Pair is created:

GEQ(ns(n0), ns(Y')) -> GEQ(n0, Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 51
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
three new Dependency Pairs are created:

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 52
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
two new Dependency Pairs are created:

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 53
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
two new Dependency Pairs are created:

GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 54
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
one new Dependency Pair is created:

GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 55
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
one new Dependency Pair is created:

GEQ(ns(X''), ns(n0)) -> GEQ(X'', n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 56
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(X''), ns(ns(X'))) -> GEQ(X'', s(X'))
one new Dependency Pair is created:

GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 57
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(n0), ns(n0)) -> GEQ(0, 0)
two new Dependency Pairs are created:

GEQ(ns(n0), ns(n0)) -> GEQ(n0, 0)
GEQ(ns(n0), ns(n0)) -> GEQ(0, n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 58
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), 0)
two new Dependency Pairs are created:

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X'')), ns(n0)) -> GEQ(s(X''), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 59
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(X''), ns(n0)) -> GEQ(X'', 0)
one new Dependency Pair is created:

GEQ(ns(X''), ns(n0)) -> GEQ(X'', n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 60
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, s(X''))
two new Dependency Pairs are created:

GEQ(ns(n0), ns(ns(X''))) -> GEQ(n0, s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 61
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(s(X'''), s(X''))
two new Dependency Pairs are created:

GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 62
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(X0), ns(ns(X''))) -> GEQ(X0, s(X''))
one new Dependency Pair is created:

GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 63
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
three new Dependency Pairs are created:

GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 64
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(n0), ns(Y')) -> GEQ(0, Y')
one new Dependency Pair is created:

GEQ(ns(n0), ns(Y')) -> GEQ(n0, Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 65
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X'')), ns(Y')) -> GEQ(s(X''), Y')
one new Dependency Pair is created:

GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 66
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(n0), ns(ns(X''))) -> GEQ(0, ns(X''))
one new Dependency Pair is created:

GEQ(ns(n0), ns(ns(X''))) -> GEQ(n0, ns(X''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 67
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
one new Dependency Pair is created:

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 68
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
one new Dependency Pair is created:

GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 69
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
one new Dependency Pair is created:

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 70
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X''')), ns(ns(X'))) -> GEQ(ns(X'''), s(X'))
one new Dependency Pair is created:

GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 71
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
one new Dependency Pair is created:

GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 72
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), 0)
one new Dependency Pair is created:

GEQ(ns(ns(X''')), ns(n0)) -> GEQ(ns(X'''), n0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 73
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
one new Dependency Pair is created:

GEQ(ns(n0), ns(ns(X'''))) -> GEQ(n0, ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 74
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X'''')), ns(ns(X''))) -> GEQ(ns(X''''), s(X''))
one new Dependency Pair is created:

GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 75
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X''')), ns(ns(X''''))) -> GEQ(s(X'''), ns(X''''))
one new Dependency Pair is created:

GEQ(ns(ns(X''''')), ns(ns(X''''))) -> GEQ(ns(X'''''), ns(X''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 76
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X''''')), ns(ns(X''''))) -> GEQ(ns(X'''''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(n0), ns(ns(X'''))) -> GEQ(0, ns(X'''))
one new Dependency Pair is created:

GEQ(ns(n0), ns(ns(X'''))) -> GEQ(n0, ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 77
                 ↳Narrowing Transformation
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))
GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X''''')), ns(ns(X''''))) -> GEQ(ns(X'''''), ns(X''''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

GEQ(ns(ns(X'')), ns(ns(X'''))) -> GEQ(s(X''), ns(X'''))
one new Dependency Pair is created:

GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 78
                 ↳Argument Filtering and Ordering
       →DP Problem 3
         ↳Remaining


Dependency Pairs:

GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))
GEQ(ns(ns(X''''')), ns(ns(X''''))) -> GEQ(ns(X'''''), ns(X''''))
GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')
GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





The following dependency pairs can be strictly oriented:

GEQ(ns(ns(X'''')), ns(ns(X'''))) -> GEQ(ns(X''''), ns(X'''))
GEQ(ns(ns(X''''')), ns(ns(X''''))) -> GEQ(ns(X'''''), ns(X''''))
GEQ(ns(ns(X''')), ns(ns(X''))) -> GEQ(ns(X'''), ns(X''))
GEQ(ns(ns(X''')), ns(Y')) -> GEQ(ns(X'''), Y')
GEQ(ns(X''), ns(ns(X'''))) -> GEQ(X'', ns(X'''))
GEQ(ns(X0), ns(ns(X'''))) -> GEQ(X0, ns(X'''))
GEQ(ns(X''), ns(Y')) -> GEQ(X'', Y')


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
GEQ(x1, x2) -> GEQ(x1, x2)
ns(x1) -> ns(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 42
             ↳Nar
             ...
               →DP Problem 79
                 ↳Dependency Graph
       →DP Problem 3
         ↳Remaining


Dependency Pair:


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
       →DP Problem 3
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

DIV(s(X), ns(Y)) -> DIV(minus(X, activate(Y)), ns(activate(Y)))


Rules:


minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X




Termination of R could not be shown.
Duration:
0:10 minutes