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:

not(x) → xor(x, true)
implies(x, y) → xor(and(x, y), xor(x, true))
or(x, y) → xor(and(x, y), xor(x, y))
=(x, y) → xor(x, xor(y, true))

Q is empty.


↳ QTRS
  ↳ RRRPoloQTRSProof

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

not(x) → xor(x, true)
implies(x, y) → xor(and(x, y), xor(x, true))
or(x, y) → xor(and(x, y), xor(x, y))
=(x, y) → xor(x, xor(y, true))

Q is empty.

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

not(x) → xor(x, true)
implies(x, y) → xor(and(x, y), xor(x, true))
or(x, y) → xor(and(x, y), xor(x, y))
=(x, y) → xor(x, xor(y, true))

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

not(x) → xor(x, true)
or(x, y) → xor(and(x, y), xor(x, y))
=(x, y) → xor(x, xor(y, true))
Used ordering:
Polynomial interpretation [25]:

POL(=(x1, x2)) = 2 + x1 + x2   
POL(and(x1, x2)) = 1 + x1 + x2   
POL(implies(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(not(x1)) = 2 + 2·x1   
POL(or(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(true) = 1   
POL(xor(x1, x2)) = x1 + x2   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

implies(x, y) → xor(and(x, y), xor(x, true))

Q is empty.

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

implies(x, y) → xor(and(x, y), xor(x, true))

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

implies(x, y) → xor(and(x, y), xor(x, true))
Used ordering:
Polynomial interpretation [25]:

POL(and(x1, x2)) = x1 + x2   
POL(implies(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(true) = 0   
POL(xor(x1, x2)) = x1 + x2   




↳ 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.