Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(c(a(x1)))) → b(a(c(b(a(b(x1))))))
a(d(x1)) → c(x1)
a(f(f(x1))) → g(x1)
b(g(x1)) → g(b(x1))
c(x1) → f(f(x1))
c(a(c(x1))) → b(c(a(b(c(x1)))))
c(d(x1)) → a(a(x1))
g(x1) → c(a(x1))
g(x1) → d(d(d(d(x1))))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(c(a(x1)))) → b(a(c(b(a(b(x1))))))
a(d(x1)) → c(x1)
a(f(f(x1))) → g(x1)
b(g(x1)) → g(b(x1))
c(x1) → f(f(x1))
c(a(c(x1))) → b(c(a(b(c(x1)))))
c(d(x1)) → a(a(x1))
g(x1) → c(a(x1))
g(x1) → d(d(d(d(x1))))
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:
C(a(c(x1))) → A(b(c(x1)))
C(d(x1)) → A(a(x1))
A(b(c(a(x1)))) → B(a(c(b(a(b(x1))))))
C(a(c(x1))) → C(a(b(c(x1))))
A(d(x1)) → C(x1)
G(x1) → C(a(x1))
A(b(c(a(x1)))) → A(b(x1))
C(d(x1)) → A(x1)
B(g(x1)) → G(b(x1))
A(b(c(a(x1)))) → A(c(b(a(b(x1)))))
A(f(f(x1))) → G(x1)
G(x1) → A(x1)
A(b(c(a(x1)))) → B(x1)
C(a(c(x1))) → B(c(x1))
A(b(c(a(x1)))) → B(a(b(x1)))
A(b(c(a(x1)))) → C(b(a(b(x1))))
C(a(c(x1))) → B(c(a(b(c(x1)))))
B(g(x1)) → B(x1)
The TRS R consists of the following rules:
a(b(c(a(x1)))) → b(a(c(b(a(b(x1))))))
a(d(x1)) → c(x1)
a(f(f(x1))) → g(x1)
b(g(x1)) → g(b(x1))
c(x1) → f(f(x1))
c(a(c(x1))) → b(c(a(b(c(x1)))))
c(d(x1)) → a(a(x1))
g(x1) → c(a(x1))
g(x1) → d(d(d(d(x1))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
C(a(c(x1))) → A(b(c(x1)))
C(d(x1)) → A(a(x1))
A(b(c(a(x1)))) → B(a(c(b(a(b(x1))))))
C(a(c(x1))) → C(a(b(c(x1))))
A(d(x1)) → C(x1)
G(x1) → C(a(x1))
A(b(c(a(x1)))) → A(b(x1))
C(d(x1)) → A(x1)
B(g(x1)) → G(b(x1))
A(b(c(a(x1)))) → A(c(b(a(b(x1)))))
A(f(f(x1))) → G(x1)
G(x1) → A(x1)
A(b(c(a(x1)))) → B(x1)
C(a(c(x1))) → B(c(x1))
A(b(c(a(x1)))) → B(a(b(x1)))
A(b(c(a(x1)))) → C(b(a(b(x1))))
C(a(c(x1))) → B(c(a(b(c(x1)))))
B(g(x1)) → B(x1)
The TRS R consists of the following rules:
a(b(c(a(x1)))) → b(a(c(b(a(b(x1))))))
a(d(x1)) → c(x1)
a(f(f(x1))) → g(x1)
b(g(x1)) → g(b(x1))
c(x1) → f(f(x1))
c(a(c(x1))) → b(c(a(b(c(x1)))))
c(d(x1)) → a(a(x1))
g(x1) → c(a(x1))
g(x1) → d(d(d(d(x1))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.