Termination w.r.t. Q proof of /tmp/tmpGn0Pe

Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

backslash(x, x) → e
/(x, x) → e
.(e, x) → x
.(x, e) → x
backslash(e, x) → x
/(x, e) → x
.(x, backslash(x, y)) → y
.(/(y, x), x) → y
backslash(x, .(x, y)) → y
/(.(y, x), x) → y
/(x, backslash(y, x)) → y
backslash(/(x, y), x) → y

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

backslash(x, x) → e
/(x, x) → e
.(e, x) → x
.(x, e) → x
backslash(e, x) → x
/(x, e) → x
.(x, backslash(x, y)) → y
.(/(y, x), x) → y
backslash(x, .(x, y)) → y
/(.(y, x), x) → y
/(x, backslash(y, x)) → y
backslash(/(x, y), x) → y

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

backslash(x, x) → e
/(x, x) → e
.(e, x) → x
.(x, e) → x
backslash(e, x) → x
/(x, e) → x
.(x, backslash(x, y)) → y
.(/(y, x), x) → y
backslash(x, .(x, y)) → y
/(.(y, x), x) → y
/(x, backslash(y, x)) → y
backslash(/(x, y), x) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

backslash(x, x) → e
/(x, x) → e
.(e, x) → x
.(x, e) → x
backslash(e, x) → x
/(x, e) → x
.(x, backslash(x, y)) → y
.(/(y, x), x) → y
backslash(x, .(x, y)) → y
/(.(y, x), x) → y
/(x, backslash(y, x)) → y
backslash(/(x, y), x) → y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.