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
  ↳ RRRPoloQTRSProof

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.

The following Q TRS is given: 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.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

.(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
Used ordering:
Polynomial interpretation [25]:

POL(.(x1, x2)) = 1 + x1 + 2·x2   
POL(/(x1, x2)) = 1 + x1 + x2   
POL(backslash(x1, x2)) = 1 + x1 + x2   
POL(e) = 1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

backslash(x, x) → e
/(x, x) → e

Q is empty.

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

backslash(x, x) → e
/(x, x) → e

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

backslash(x, x) → e
/(x, x) → e
Used ordering:
Polynomial interpretation [25]:

POL(/(x1, x2)) = 2 + x1 + x2   
POL(backslash(x1, x2)) = 2 + x1 + x2   
POL(e) = 1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RisEmptyProof

Q restricted rewrite system:
R is empty.
Q is empty.

The TRS R is empty. Hence, termination is trivially proven.