R
↳Dependency Pair Analysis
2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons(X, Z)) -> ACTIVATE(Z)
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> ACTIVATE(Z)
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSNEG(s(N), cons(X, Z)) -> ACTIVATE(Z)
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> ACTIVATE(Z)
PI(X) -> 2NDSPOS(X, from(0))
PI(X) -> FROM(0)
PLUS(s(X), Y) -> S(plus(X, Y))
PLUS(s(X), Y) -> PLUS(X, Y)
TIMES(s(X), Y) -> PLUS(Y, times(X, Y))
TIMES(s(X), Y) -> TIMES(X, Y)
SQUARE(X) -> TIMES(X, X)
ACTIVATE(nfrom(X)) -> FROM(activate(X))
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(ns(X)) -> S(activate(X))
ACTIVATE(ns(X)) -> ACTIVATE(X)
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳AFS
→DP Problem 3
↳Nar
→DP Problem 4
↳Remaining
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
ACTIVATE(ns(X)) -> ACTIVATE(X)
ACTIVATE(nfrom(X)) -> ACTIVATE(X)
ACTIVATE(x1) -> ACTIVATE(x1)
ns(x1) -> ns(x1)
nfrom(x1) -> nfrom(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 5
↳Dependency Graph
→DP Problem 2
↳AFS
→DP Problem 3
↳Nar
→DP Problem 4
↳Remaining
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Argument Filtering and Ordering
→DP Problem 3
↳Nar
→DP Problem 4
↳Remaining
PLUS(s(X), Y) -> PLUS(X, Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
PLUS(s(X), Y) -> PLUS(X, Y)
PLUS(x1, x2) -> PLUS(x1, x2)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 6
↳Dependency Graph
→DP Problem 3
↳Nar
→DP Problem 4
↳Remaining
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳Narrowing Transformation
→DP Problem 4
↳Remaining
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
four new Dependency Pairs are created:
2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N'), cons(X, Z)) -> 2NDSPOS(ns(N'), cons2(X, activate(Z)))
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons(X, ns(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳Nar
→DP Problem 7
↳Narrowing Transformation
→DP Problem 4
↳Remaining
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons(X, ns(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
three new Dependency Pairs are created:
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, ns(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳Nar
→DP Problem 7
↳Nar
...
→DP Problem 8
↳Narrowing Transformation
→DP Problem 4
↳Remaining
2NDSPOS(s(N), cons(X, ns(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons2(X, cons(Y, ns(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
four new Dependency Pairs are created:
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSNEG(s(N'), cons(X, Z)) -> 2NDSNEG(ns(N'), cons2(X, activate(Z)))
2NDSNEG(s(N), cons(X, nfrom(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons(X, ns(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳Nar
→DP Problem 7
↳Nar
...
→DP Problem 9
↳Narrowing Transformation
→DP Problem 4
↳Remaining
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons(X, ns(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSNEG(s(N), cons(X, nfrom(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, ns(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons(X, ns(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
three new Dependency Pairs are created:
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons2(X, cons(Y, ns(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳Nar
→DP Problem 4
↳Remaining Obligation(s)
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, ns(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, ns(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons(X, ns(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, ns(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons(X, nfrom(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
TIMES(s(X), Y) -> TIMES(X, Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 3
↳Nar
→DP Problem 4
↳Remaining Obligation(s)
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, ns(X''))) -> 2NDSPOS(s(N), cons2(X, s(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, ns(X'')))) -> 2NDSPOS(N, s(activate(X'')))
2NDSNEG(s(N), cons(X, ns(X''))) -> 2NDSNEG(s(N), cons2(X, s(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, ns(X'')))) -> 2NDSNEG(N, s(activate(X'')))
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(activate(X''))))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSPOS(N, from(activate(X'')))
2NDSNEG(s(N), cons(X, nfrom(X''))) -> 2NDSNEG(s(N), cons2(X, from(activate(X''))))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(activate(X'')))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X
TIMES(s(X), Y) -> TIMES(X, Y)
from(X) -> cons(X, nfrom(ns(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
s(X) -> ns(X)
activate(nfrom(X)) -> from(activate(X))
activate(ns(X)) -> s(activate(X))
activate(X) -> X