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:
f(n__f(n__a)) → f(n__g(f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(X)
activate(X) → X
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(n__f(n__a)) → f(n__g(f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(X)
activate(X) → X
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
F(n__f(n__a)) → F(n__g(f(n__a)))
ACTIVATE(n__f(X)) → F(X)
ACTIVATE(n__a) → A
F(n__f(n__a)) → F(n__a)
ACTIVATE(n__g(X)) → G(X)
The TRS R consists of the following rules:
f(n__f(n__a)) → f(n__g(f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(X)
activate(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:
F(n__f(n__a)) → F(n__g(f(n__a)))
ACTIVATE(n__f(X)) → F(X)
ACTIVATE(n__a) → A
F(n__f(n__a)) → F(n__a)
ACTIVATE(n__g(X)) → G(X)
The TRS R consists of the following rules:
f(n__f(n__a)) → f(n__g(f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(X)
activate(X) → X
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 5 less nodes.