Termination w.r.t. Q of the following Term Rewriting System could be proven:
Q restricted rewrite system:
The TRS R consists of the following rules:
f1(0) -> cons2(0, n__f1(n__s1(n__0)))
f1(s1(0)) -> f1(p1(s1(0)))
p1(s1(0)) -> 0
f1(X) -> n__f1(X)
s1(X) -> n__s1(X)
0 -> n__0
activate1(n__f1(X)) -> f1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__0) -> 0
activate1(X) -> X
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f1(0) -> cons2(0, n__f1(n__s1(n__0)))
f1(s1(0)) -> f1(p1(s1(0)))
p1(s1(0)) -> 0
f1(X) -> n__f1(X)
s1(X) -> n__s1(X)
0 -> n__0
activate1(n__f1(X)) -> f1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__0) -> 0
activate1(X) -> X
Q is empty.
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE1(n__s1(X)) -> S1(activate1(X))
ACTIVATE1(n__0) -> 01
ACTIVATE1(n__f1(X)) -> F1(activate1(X))
F1(s1(0)) -> P1(s1(0))
F1(s1(0)) -> F1(p1(s1(0)))
ACTIVATE1(n__f1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)
The TRS R consists of the following rules:
f1(0) -> cons2(0, n__f1(n__s1(n__0)))
f1(s1(0)) -> f1(p1(s1(0)))
p1(s1(0)) -> 0
f1(X) -> n__f1(X)
s1(X) -> n__s1(X)
0 -> n__0
activate1(n__f1(X)) -> f1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__0) -> 0
activate1(X) -> X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE1(n__s1(X)) -> S1(activate1(X))
ACTIVATE1(n__0) -> 01
ACTIVATE1(n__f1(X)) -> F1(activate1(X))
F1(s1(0)) -> P1(s1(0))
F1(s1(0)) -> F1(p1(s1(0)))
ACTIVATE1(n__f1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)
The TRS R consists of the following rules:
f1(0) -> cons2(0, n__f1(n__s1(n__0)))
f1(s1(0)) -> f1(p1(s1(0)))
p1(s1(0)) -> 0
f1(X) -> n__f1(X)
s1(X) -> n__s1(X)
0 -> n__0
activate1(n__f1(X)) -> f1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__0) -> 0
activate1(X) -> X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 2 SCCs with 4 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F1(s1(0)) -> F1(p1(s1(0)))
The TRS R consists of the following rules:
f1(0) -> cons2(0, n__f1(n__s1(n__0)))
f1(s1(0)) -> f1(p1(s1(0)))
p1(s1(0)) -> 0
f1(X) -> n__f1(X)
s1(X) -> n__s1(X)
0 -> n__0
activate1(n__f1(X)) -> f1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__0) -> 0
activate1(X) -> X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
F1(s1(0)) -> F1(p1(s1(0)))
Used argument filtering: F1(x1) = x1
s1(x1) = s
0 = 0
p1(x1) = p
n__0 = n__0
n__s1(x1) = n__s
Used ordering: Quasi Precedence:
s > [p, 0] > n__0
s > n__s
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
↳ QDP
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f1(0) -> cons2(0, n__f1(n__s1(n__0)))
f1(s1(0)) -> f1(p1(s1(0)))
p1(s1(0)) -> 0
f1(X) -> n__f1(X)
s1(X) -> n__s1(X)
0 -> n__0
activate1(n__f1(X)) -> f1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__0) -> 0
activate1(X) -> X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE1(n__f1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)
The TRS R consists of the following rules:
f1(0) -> cons2(0, n__f1(n__s1(n__0)))
f1(s1(0)) -> f1(p1(s1(0)))
p1(s1(0)) -> 0
f1(X) -> n__f1(X)
s1(X) -> n__s1(X)
0 -> n__0
activate1(n__f1(X)) -> f1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__0) -> 0
activate1(X) -> X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
ACTIVATE1(n__s1(X)) -> ACTIVATE1(X)
Used argument filtering: ACTIVATE1(x1) = x1
n__f1(x1) = x1
n__s1(x1) = n__s1(x1)
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ QDPAfsSolverProof
Q DP problem:
The TRS P consists of the following rules:
ACTIVATE1(n__f1(X)) -> ACTIVATE1(X)
The TRS R consists of the following rules:
f1(0) -> cons2(0, n__f1(n__s1(n__0)))
f1(s1(0)) -> f1(p1(s1(0)))
p1(s1(0)) -> 0
f1(X) -> n__f1(X)
s1(X) -> n__s1(X)
0 -> n__0
activate1(n__f1(X)) -> f1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__0) -> 0
activate1(X) -> X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.
ACTIVATE1(n__f1(X)) -> ACTIVATE1(X)
Used argument filtering: ACTIVATE1(x1) = x1
n__f1(x1) = n__f1(x1)
Used ordering: Quasi Precedence:
trivial
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f1(0) -> cons2(0, n__f1(n__s1(n__0)))
f1(s1(0)) -> f1(p1(s1(0)))
p1(s1(0)) -> 0
f1(X) -> n__f1(X)
s1(X) -> n__s1(X)
0 -> n__0
activate1(n__f1(X)) -> f1(activate1(X))
activate1(n__s1(X)) -> s1(activate1(X))
activate1(n__0) -> 0
activate1(X) -> X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.