(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X) → cons(X, f(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, Z)
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(X) → cons(X, f(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, Z)
The set Q consists of the following terms:
f(x0)
g(0)
g(s(x0))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(X) → F(g(X))
F(X) → G(X)
G(s(X)) → G(X)
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
The TRS R consists of the following rules:
f(X) → cons(X, f(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, Z)
The set Q consists of the following terms:
f(x0)
g(0)
g(s(x0))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 1 less node.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
The TRS R consists of the following rules:
f(X) → cons(X, f(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, Z)
The set Q consists of the following terms:
f(x0)
g(0)
g(s(x0))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
We have to consider all minimal (P,Q,R)-chains.
(8) 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, f(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, Z)
The set Q consists of the following terms:
f(x0)
g(0)
g(s(x0))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
We have to consider all minimal (P,Q,R)-chains.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(X) → F(g(X))
The TRS R consists of the following rules:
f(X) → cons(X, f(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, Z)
The set Q consists of the following terms:
f(x0)
g(0)
g(s(x0))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
We have to consider all minimal (P,Q,R)-chains.