(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(X)) → c(n__f(g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
d(X) → n__d(X)
activate(n__f(X)) → f(X)
activate(n__d(X)) → d(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:
F(f(X)) → C(n__f(g(n__f(X))))
C(X) → D(activate(X))
C(X) → ACTIVATE(X)
H(X) → C(n__d(X))
ACTIVATE(n__f(X)) → F(X)
ACTIVATE(n__d(X)) → D(X)
The TRS R consists of the following rules:
f(f(X)) → c(n__f(g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
d(X) → n__d(X)
activate(n__f(X)) → f(X)
activate(n__d(X)) → d(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 1 SCC with 3 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(X) → ACTIVATE(X)
ACTIVATE(n__f(X)) → F(X)
F(f(X)) → C(n__f(g(n__f(X))))
The TRS R consists of the following rules:
f(f(X)) → c(n__f(g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
d(X) → n__d(X)
activate(n__f(X)) → f(X)
activate(n__d(X)) → d(X)
activate(X) → 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.
C(X) → ACTIVATE(X)
ACTIVATE(n__f(X)) → F(X)
F(f(X)) → C(n__f(g(n__f(X))))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
C(
x0,
x1) =
C(
x1)
ACTIVATE(
x0,
x1) =
ACTIVATE(
x1)
F(
x0,
x1) =
F(
x1)
Tags:
C has argument tags [0,2] and root tag 3
ACTIVATE has argument tags [0,2] and root tag 2
F has argument tags [0,2] and root tag 1
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
C(
x1) =
x1
ACTIVATE(
x1) =
ACTIVATE
n__f(
x1) =
x1
F(
x1) =
F
f(
x1) =
f
g(
x1) =
g
Recursive path order with status [RPO].
Quasi-Precedence:
f > [F, g]
Status:
ACTIVATE: multiset
F: multiset
f: multiset
g: multiset
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(f(X)) → c(n__f(g(n__f(X))))
c(X) → d(activate(X))
h(X) → c(n__d(X))
f(X) → n__f(X)
d(X) → n__d(X)
activate(n__f(X)) → f(X)
activate(n__d(X)) → d(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE