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:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
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:
F(a, f(a, f(a, f(x, b)))) → F(f(a, f(a, f(a, x))), b)
F(f(f(a, x), b), b) → F(f(x, b), b)
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, f(a, x)))
F(f(f(a, x), b), b) → F(x, b)
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, x))
F(f(f(a, x), b), b) → F(f(a, f(f(x, b), b)), b)
F(f(f(a, x), b), b) → F(a, f(f(x, b), b))
F(a, f(a, f(a, f(x, b)))) → F(a, x)
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
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:
F(a, f(a, f(a, f(x, b)))) → F(f(a, f(a, f(a, x))), b)
F(f(f(a, x), b), b) → F(f(x, b), b)
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, f(a, x)))
F(f(f(a, x), b), b) → F(x, b)
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, x))
F(f(f(a, x), b), b) → F(f(a, f(f(x, b), b)), b)
F(f(f(a, x), b), b) → F(a, f(f(x, b), b))
F(a, f(a, f(a, f(x, b)))) → F(a, x)
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
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
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
F(a, f(a, f(a, f(x, b)))) → F(f(a, f(a, f(a, x))), b)
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, f(a, x)))
F(f(f(a, x), b), b) → F(f(x, b), b)
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, x))
F(f(f(a, x), b), b) → F(x, b)
F(f(f(a, x), b), b) → F(f(a, f(f(x, b), b)), b)
F(f(f(a, x), b), b) → F(a, f(f(x, b), b))
F(a, f(a, f(a, f(x, b)))) → F(a, x)
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 2 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(f(f(a, x), b), b) → F(f(x, b), b)
F(f(f(a, x), b), b) → F(x, b)
F(f(f(a, x), b), b) → F(f(a, f(f(x, b), b)), b)
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, f(a, x)))
F(a, f(a, f(a, f(x, b)))) → F(a, f(a, x))
F(a, f(a, f(a, f(x, b)))) → F(a, x)
The TRS R consists of the following rules:
f(a, f(a, f(a, f(x, b)))) → f(f(a, f(a, f(a, x))), b)
f(f(f(a, x), b), b) → f(f(a, f(f(x, b), b)), b)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.