Termination Proof using AProVE

Term Rewriting System R:
[X, Z, N, Y, X1, X2]
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FROM(X) -> CONS(X, n_{_from}(n_{_s}(X)))
2NDSPOS(s(N), cons(X, n_{_cons}(Y, Z))) -> ACTIVATE(Y)
2NDSPOS(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons(X, n_{_cons}(Y, Z))) -> ACTIVATE(Z)
2NDSNEG(s(N), cons(X, n_{_cons}(Y, Z))) -> ACTIVATE(Y)
2NDSNEG(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, n_{_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(n_{_from}(X)) -> FROM(activate(X))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_s}(X)) -> S(activate(X))
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_cons}(X1, X2)) -> CONS(activate(X1), X2)
ACTIVATE(n_{_cons}(X1, X2)) -> ACTIVATE(X1)

Furthermore, R contains four SCCs.

     ↳DPs

       →DP Problem 1

         ↳Polynomial Ordering

       →DP Problem 2

         ↳Remaining

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pairs:

ACTIVATE(n_{_cons}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)

Additionally, the following rules can be oriented:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 1 + x₁

POL(activate(x₁)) = x₁

POL(plus(x₁, x₂)) = x₂

POL(square(x₁)) = 0

POL(negrecip(x₁)) = 0

POL(posrecip(x₁)) = 0

POL(n__s(x₁)) = x₁

POL(rcons(x₁, x₂)) = 0

POL(ACTIVATE(x₁)) = x₁

POL(2ndspos(x₁, x₂)) = 0

POL(2ndsneg(x₁, x₂)) = 0

POL(n__from(x₁)) = 1 + x₁

POL(n__cons(x₁, x₂)) = x₁

POL(rnil) = 0

POL(0) = 0

POL(cons(x₁, x₂)) = x₁

POL(times(x₁, x₂)) = 0

POL(s(x₁)) = x₁

POL(pi(x₁)) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

           →DP Problem 5

             ↳Polynomial Ordering

       →DP Problem 2

         ↳Remaining

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pairs:

ACTIVATE(n_{_cons}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

The following dependency pair can be strictly oriented:

ACTIVATE(n_{_cons}(X1, X2)) -> ACTIVATE(X1)

Additionally, the following rules can be oriented:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
s(X) -> n_{_s}(X)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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)

Used ordering: Polynomial ordering with Polynomial interpretation:

POL(from(x₁)) = 1 + x₁

POL(activate(x₁)) = x₁

POL(plus(x₁, x₂)) = x₂

POL(square(x₁)) = 0

POL(negrecip(x₁)) = 0

POL(posrecip(x₁)) = 0

POL(n__s(x₁)) = x₁

POL(rcons(x₁, x₂)) = 0

POL(ACTIVATE(x₁)) = x₁

POL(2ndspos(x₁, x₂)) = 0

POL(2ndsneg(x₁, x₂)) = 0

POL(n__from(x₁)) = 1 + x₁

POL(n__cons(x₁, x₂)) = 1 + x₁

POL(rnil) = 0

POL(0) = 0

POL(cons(x₁, x₂)) = 1 + x₁

POL(times(x₁, x₂)) = 0

POL(s(x₁)) = x₁

POL(pi(x₁)) = 0

resulting in one new DP problem.

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pair:
PLUS(s(X), Y) -> PLUS(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pairs:
2NDSNEG(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSPOS(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSNEG(N, activate(Z))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pair:
TIMES(s(X), Y) -> TIMES(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pair:
PLUS(s(X), Y) -> PLUS(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pairs:
2NDSNEG(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSPOS(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSNEG(N, activate(Z))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pair:
TIMES(s(X), Y) -> TIMES(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pair:
PLUS(s(X), Y) -> PLUS(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pairs:
2NDSNEG(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSPOS(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSNEG(N, activate(Z))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pair:
TIMES(s(X), Y) -> TIMES(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Polo

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
ACTIVATE(n_{_s}(X)) -> ACTIVATE(X)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pair:
PLUS(s(X), Y) -> PLUS(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pairs:
2NDSNEG(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSPOS(s(N), cons(X, n_{_cons}(Y, Z))) -> 2NDSNEG(N, activate(Z))

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X
Dependency Pair:
TIMES(s(X), Y) -> TIMES(X, Y)

Rules:

from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, n_{_cons}(Y, Z))) -> rcons(negrecip(activate(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) -> n_{_s}(X)
cons(X1, X2) -> n_{_cons}(X1, X2)
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(activate(X))
activate(n_{_cons}(X1, X2)) -> cons(activate(X1), X2)
activate(X) -> X

Termination of R could not be shown.
Duration:
0:04 minutes

POL(from(x₁))	= 1 + x₁
POL(activate(x₁))	= x₁
POL(plus(x₁, x₂))	= x₂
POL(square(x₁))	= 0
POL(negrecip(x₁))	= 0
POL(posrecip(x₁))	= 0
POL(n__s(x₁))	= x₁
POL(rcons(x₁, x₂))	= 0
POL(ACTIVATE(x₁))	= x₁
POL(2ndspos(x₁, x₂))	= 0
POL(2ndsneg(x₁, x₂))	= 0
POL(n__from(x₁))	= 1 + x₁
POL(n__cons(x₁, x₂))	= x₁
POL(rnil)	= 0
POL(0)	= 0
POL(cons(x₁, x₂))	= x₁
POL(times(x₁, x₂))	= 0
POL(s(x₁))	= x₁
POL(pi(x₁))	= 0