Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z]
f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

G(s(X)) -> G(X)
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_f}(X)) -> F(activate(X))
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_g}(X)) -> G(activate(X))
ACTIVATE(n_{_g}(X)) -> ACTIVATE(X)

Furthermore, R contains three SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
G(s(X)) -> G(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X
Dependency Pairs:
ACTIVATE(n_{_g}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X
Dependency Pair:
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
G(s(X)) -> G(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X
Dependency Pairs:
ACTIVATE(n_{_g}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X
Dependency Pair:
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
G(s(X)) -> G(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X
Dependency Pairs:
ACTIVATE(n_{_g}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_f}(X)) -> ACTIVATE(X)

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X
Dependency Pair:
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))

Rules:

f(X) -> cons(X, n_{_f}(n_{_g}(X)))
f(X) -> n_{_f}(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> n_{_g}(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(n_{_f}(X)) -> f(activate(X))
activate(n_{_g}(X)) -> g(activate(X))
activate(X) -> X

Termination of R could not be shown.
Duration:
0:00 minutes