(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(X))
mark(c) → active(c)
mark(d) → active(d)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(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:
ACTIVE(g(X)) → MARK(h(X))
ACTIVE(g(X)) → H(X)
ACTIVE(c) → MARK(d)
ACTIVE(h(d)) → MARK(g(c))
ACTIVE(h(d)) → G(c)
MARK(g(X)) → ACTIVE(g(X))
MARK(h(X)) → ACTIVE(h(X))
MARK(c) → ACTIVE(c)
MARK(d) → ACTIVE(d)
G(mark(X)) → G(X)
G(active(X)) → G(X)
H(mark(X)) → H(X)
H(active(X)) → H(X)
The TRS R consists of the following rules:
active(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(X))
mark(c) → active(c)
mark(d) → active(d)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(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 5 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
H(active(X)) → H(X)
H(mark(X)) → H(X)
The TRS R consists of the following rules:
active(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(X))
mark(c) → active(c)
mark(d) → active(d)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(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:
G(active(X)) → G(X)
G(mark(X)) → G(X)
The TRS R consists of the following rules:
active(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(X))
mark(c) → active(c)
mark(d) → active(d)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(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:
MARK(g(X)) → ACTIVE(g(X))
ACTIVE(g(X)) → MARK(h(X))
MARK(h(X)) → ACTIVE(h(X))
ACTIVE(h(d)) → MARK(g(c))
The TRS R consists of the following rules:
active(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(X))
mark(c) → active(c)
mark(d) → active(d)
g(mark(X)) → g(X)
g(active(X)) → g(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.