(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) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
f(x1) = f(x1)
g(x1) = x1
n__f(x1) = x1
From the DPs we obtained the following set of size-change graphs:
- ACTIVATE(n__f(X)) → F(X) (allowed arguments on rhs = {1})
The graph contains the following edges 1 >= 1
- F(f(X)) → C(n__f(g(n__f(X)))) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
- C(X) → ACTIVATE(X) (allowed arguments on rhs = {1})
The graph contains the following edges 1 >= 1
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(6) TRUE