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(x, x) → f(i(x), g(g(x)))
f(x, y) → x
g(x) → i(x)
f(x, i(x)) → f(x, x)
f(i(x), i(g(x))) → a
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, x) → f(i(x), g(g(x)))
f(x, y) → x
g(x) → i(x)
f(x, i(x)) → f(x, x)
f(i(x), i(g(x))) → a
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(x, x) → F(i(x), g(g(x)))
F(x, x) → G(x)
F(x, i(x)) → F(x, x)
F(x, x) → G(g(x))
The TRS R consists of the following rules:
f(x, x) → f(i(x), g(g(x)))
f(x, y) → x
g(x) → i(x)
f(x, i(x)) → f(x, x)
f(i(x), i(g(x))) → a
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(x, x) → F(i(x), g(g(x)))
F(x, x) → G(x)
F(x, i(x)) → F(x, x)
F(x, x) → G(g(x))
The TRS R consists of the following rules:
f(x, x) → f(i(x), g(g(x)))
f(x, y) → x
g(x) → i(x)
f(x, i(x)) → f(x, x)
f(i(x), i(g(x))) → a
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(x, x) → F(i(x), g(g(x)))
F(x, x) → G(x)
F(x, i(x)) → F(x, x)
F(x, x) → G(g(x))
The TRS R consists of the following rules:
f(x, x) → f(i(x), g(g(x)))
f(x, y) → x
g(x) → i(x)
f(x, i(x)) → f(x, x)
f(i(x), i(g(x))) → a
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 2 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(x, x) → F(i(x), g(g(x)))
F(x, i(x)) → F(x, x)
The TRS R consists of the following rules:
f(x, x) → f(i(x), g(g(x)))
f(x, y) → x
g(x) → i(x)
f(x, i(x)) → f(x, x)
f(i(x), i(g(x))) → a
Q is empty.
We have to consider all minimal (P,Q,R)-chains.