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(f(f(a, x), y), z) → f(f(x, z), f(y, z))
f(f(b, x), y) → x
f(c, y) → y
Q is empty.
↳ QTRS
↳ Overlay + Local Confluence
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(f(a, x), y), z) → f(f(x, z), f(y, z))
f(f(b, x), y) → x
f(c, y) → y
Q is empty.
The TRS is overlay and locally confluent. By [15] we can switch to innermost.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(f(a, x), y), z) → f(f(x, z), f(y, z))
f(f(b, x), y) → x
f(c, y) → y
The set Q consists of the following terms:
f(f(f(a, x0), x1), x2)
f(f(b, x0), x1)
f(c, x0)
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(f(f(a, x), y), z) → F(x, z)
F(f(f(a, x), y), z) → F(f(x, z), f(y, z))
F(f(f(a, x), y), z) → F(y, z)
The TRS R consists of the following rules:
f(f(f(a, x), y), z) → f(f(x, z), f(y, z))
f(f(b, x), y) → x
f(c, y) → y
The set Q consists of the following terms:
f(f(f(a, x0), x1), x2)
f(f(b, x0), x1)
f(c, x0)
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(f(f(a, x), y), z) → F(x, z)
F(f(f(a, x), y), z) → F(f(x, z), f(y, z))
F(f(f(a, x), y), z) → F(y, z)
The TRS R consists of the following rules:
f(f(f(a, x), y), z) → f(f(x, z), f(y, z))
f(f(b, x), y) → x
f(c, y) → y
The set Q consists of the following terms:
f(f(f(a, x0), x1), x2)
f(f(b, x0), x1)
f(c, x0)
We have to consider all minimal (P,Q,R)-chains.