(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X) → cons(X, n__f(n__g(X)))
g(0) → s(0)
g(s(X)) → s(s(g(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(X) → X
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
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)
The TRS R consists of the following rules:
f(X) → cons(X, n__f(n__g(X)))
g(0) → s(0)
g(s(X)) → s(s(g(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 3 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(s(X)) → G(X)
The TRS R consists of the following rules:
f(X) → cons(X, n__f(n__g(X)))
g(0) → s(0)
g(s(X)) → s(s(g(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE(n__g(X)) → ACTIVATE(X)
ACTIVATE(n__f(X)) → ACTIVATE(X)
The TRS R consists of the following rules:
f(X) → cons(X, n__f(n__g(X)))
g(0) → s(0)
g(s(X)) → s(s(g(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SEL(s(X), cons(Y, Z)) → SEL(X, activate(Z))
The TRS R consists of the following rules:
f(X) → cons(X, n__f(n__g(X)))
g(0) → s(0)
g(s(X)) → s(s(g(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
g(X) → n__g(X)
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.