R
↳Dependency Pair Analysis
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)
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
→DP Problem 2
↳Nar
→DP Problem 3
↳Polo
MINUS(ns(X), ns(Y)) -> MINUS(activate(X), activate(Y))
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
six new Dependency Pairs are created:
MINUS(ns(X), ns(Y)) -> MINUS(activate(X), activate(Y))
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')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Narrowing Transformation
→DP Problem 2
↳Nar
→DP Problem 3
↳Polo
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))
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
four new Dependency Pairs are created:
MINUS(ns(n0), ns(Y)) -> MINUS(0, activate(Y))
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')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 5
↳Narrowing Transformation
→DP Problem 2
↳Nar
→DP Problem 3
↳Polo
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')
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
four new Dependency Pairs are created:
MINUS(ns(ns(X'')), ns(Y)) -> MINUS(s(X''), activate(Y))
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')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 6
↳Narrowing Transformation
→DP Problem 2
↳Nar
→DP Problem 3
↳Polo
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')
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
three new Dependency Pairs are created:
MINUS(ns(X''), ns(Y)) -> MINUS(X'', activate(Y))
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')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 7
↳Narrowing Transformation
→DP Problem 2
↳Nar
→DP Problem 3
↳Polo
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')
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
four new Dependency Pairs are created:
MINUS(ns(X), ns(n0)) -> MINUS(activate(X), 0)
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)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 8
↳Narrowing Transformation
→DP Problem 2
↳Nar
→DP Problem 3
↳Polo
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')
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
four new Dependency Pairs are created:
MINUS(ns(X), ns(ns(X''))) -> MINUS(activate(X), s(X''))
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'''))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 9
↳Narrowing Transformation
→DP Problem 2
↳Nar
→DP Problem 3
↳Polo
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)
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
three new Dependency Pairs are created:
MINUS(ns(X), ns(Y')) -> MINUS(activate(X), Y')
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')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 10
↳Polynomial Ordering
→DP Problem 2
↳Nar
→DP Problem 3
↳Polo
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'''))
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
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(ns(X'))) -> MINUS(X'', s(X'))
MINUS(ns(ns(X'')), ns(ns(X'))) -> MINUS(s(X''), s(X'))
MINUS(ns(ns(X''')), ns(Y)) -> MINUS(ns(X'''), activate(Y))
MINUS(ns(n0), ns(ns(X'))) -> MINUS(0, s(X'))
MINUS(ns(X), ns(ns(X'''))) -> MINUS(activate(X), ns(X'''))
s(X) -> ns(X)
0 -> n0
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X
POL(activate(x1)) = x1 POL(0) = 0 POL(MINUS(x1, x2)) = x2 POL(n__s(x1)) = 1 + x1 POL(s(x1)) = 1 + x1 POL(n__0) = 0
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 11
↳Dependency Graph
→DP Problem 2
↳Nar
→DP Problem 3
↳Polo
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
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Narrowing Transformation
→DP Problem 3
↳Polo
GEQ(ns(X), ns(Y)) -> GEQ(activate(X), activate(Y))
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
six new Dependency Pairs are created:
GEQ(ns(X), ns(Y)) -> GEQ(activate(X), activate(Y))
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')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
→DP Problem 12
↳Narrowing Transformation
→DP Problem 3
↳Polo
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))
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
four new Dependency Pairs are created:
GEQ(ns(n0), ns(Y)) -> GEQ(0, activate(Y))
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')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
→DP Problem 12
↳Nar
...
→DP Problem 13
↳Narrowing Transformation
→DP Problem 3
↳Polo
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')
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
four new Dependency Pairs are created:
GEQ(ns(ns(X'')), ns(Y)) -> GEQ(s(X''), activate(Y))
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')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
→DP Problem 12
↳Nar
...
→DP Problem 14
↳Narrowing Transformation
→DP Problem 3
↳Polo
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')
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
three new Dependency Pairs are created:
GEQ(ns(X''), ns(Y)) -> GEQ(X'', activate(Y))
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')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
→DP Problem 12
↳Nar
...
→DP Problem 15
↳Narrowing Transformation
→DP Problem 3
↳Polo
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')
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
four new Dependency Pairs are created:
GEQ(ns(X), ns(n0)) -> GEQ(activate(X), 0)
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)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
→DP Problem 12
↳Nar
...
→DP Problem 16
↳Narrowing Transformation
→DP Problem 3
↳Polo
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')
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
four new Dependency Pairs are created:
GEQ(ns(X), ns(ns(X''))) -> GEQ(activate(X), s(X''))
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'''))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
→DP Problem 12
↳Nar
...
→DP Problem 17
↳Narrowing Transformation
→DP Problem 3
↳Polo
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)
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
three new Dependency Pairs are created:
GEQ(ns(X), ns(Y')) -> GEQ(activate(X), Y')
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')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
→DP Problem 12
↳Nar
...
→DP Problem 18
↳Polynomial Ordering
→DP Problem 3
↳Polo
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'''))
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
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(ns(X'))) -> GEQ(X'', s(X'))
GEQ(ns(ns(X'')), ns(ns(X'))) -> GEQ(s(X''), s(X'))
GEQ(ns(ns(X''')), ns(Y)) -> GEQ(ns(X'''), activate(Y))
GEQ(ns(n0), ns(ns(X'))) -> GEQ(0, s(X'))
GEQ(ns(X), ns(ns(X'''))) -> GEQ(activate(X), ns(X'''))
s(X) -> ns(X)
0 -> n0
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X
POL(GEQ(x1, x2)) = x2 POL(activate(x1)) = x1 POL(0) = 0 POL(n__s(x1)) = 1 + x1 POL(s(x1)) = 1 + x1 POL(n__0) = 0
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
→DP Problem 12
↳Nar
...
→DP Problem 19
↳Dependency Graph
→DP Problem 3
↳Polo
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
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
→DP Problem 3
↳Polynomial Ordering
DIV(s(X), ns(Y)) -> DIV(minus(X, activate(Y)), ns(activate(Y)))
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
DIV(s(X), ns(Y)) -> DIV(minus(X, activate(Y)), ns(activate(Y)))
minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
0 -> n0
POL(activate(x1)) = 0 POL(0) = 0 POL(DIV(x1, x2)) = x1 POL(minus(x1, x2)) = 0 POL(n__s(x1)) = 0 POL(n__0) = 0 POL(s(x1)) = 1
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
→DP Problem 3
↳Polo
→DP Problem 20
↳Dependency Graph
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