(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(g(a)) → f(s(g(b)))
f(f(x)) → b
g(x) → f(g(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:
F(g(a)) → F(s(g(b)))
F(g(a)) → G(b)
G(x) → F(g(x))
G(x) → G(x)
The TRS R consists of the following rules:
f(g(a)) → f(s(g(b)))
f(f(x)) → b
g(x) → f(g(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 1 SCC with 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(g(a)) → G(b)
G(x) → F(g(x))
G(x) → G(x)
The TRS R consists of the following rules:
f(g(a)) → f(s(g(b)))
f(f(x)) → b
g(x) → f(g(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
F(g(a)) → G(b)
G(x) → F(g(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1) =
x1
g(
x1) =
g(
x1)
a =
a
G(
x1) =
G(
x1)
b =
b
f(
x1) =
x1
s(
x1) =
x1
Recursive Path Order [RPO].
Precedence:
a > G1 > [g1, b]
The following usable rules [FROCOS05] were oriented:
g(x) → f(g(x))
f(g(a)) → f(s(g(b)))
f(f(x)) → b
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(x) → G(x)
The TRS R consists of the following rules:
f(g(a)) → f(s(g(b)))
f(f(x)) → b
g(x) → f(g(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.