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(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
u(a(x)) → x
v(b(x)) → x
w(c(x)) → x
a(u(x)) → x
b(v(x)) → x
c(w(x)) → x
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
u(a(x)) → x
v(b(x)) → x
w(c(x)) → x
a(u(x)) → x
b(v(x)) → x
c(w(x)) → x
Q is empty.
Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
A(b(x)) → B(a(a(x)))
B(c(x)) → B(b(x))
C(a(x)) → C(x)
B(c(x)) → C(b(b(x)))
C(a(x)) → C(c(x))
A(b(x)) → A(a(x))
C(a(x)) → A(c(c(x)))
A(b(x)) → A(x)
B(c(x)) → B(x)
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
u(a(x)) → x
v(b(x)) → x
w(c(x)) → x
a(u(x)) → x
b(v(x)) → x
c(w(x)) → x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
Q DP problem:
The TRS P consists of the following rules:
A(b(x)) → B(a(a(x)))
B(c(x)) → B(b(x))
C(a(x)) → C(x)
B(c(x)) → C(b(b(x)))
C(a(x)) → C(c(x))
A(b(x)) → A(a(x))
C(a(x)) → A(c(c(x)))
A(b(x)) → A(x)
B(c(x)) → B(x)
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
u(a(x)) → x
v(b(x)) → x
w(c(x)) → x
a(u(x)) → x
b(v(x)) → x
c(w(x)) → x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
A(b(x)) → B(a(a(x)))
B(c(x)) → B(b(x))
C(a(x)) → C(x)
B(c(x)) → C(b(b(x)))
A(b(x)) → A(a(x))
C(a(x)) → C(c(x))
A(b(x)) → A(x)
C(a(x)) → A(c(c(x)))
B(c(x)) → B(x)
The TRS R consists of the following rules:
a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(b(x)))
c(a(x)) → a(c(c(x)))
u(a(x)) → x
v(b(x)) → x
w(c(x)) → x
a(u(x)) → x
b(v(x)) → x
c(w(x)) → x
Q is empty.
We have to consider all minimal (P,Q,R)-chains.