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

Q is empty.


↳ QTRS
  ↳ RRRPoloQTRSProof

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

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

Q is empty.

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

not(x) → if(x, false, true)
and(x, y) → if(x, y, false)
or(x, y) → if(x, true, y)
implies(x, y) → if(x, y, true)
=(x, x) → true
=(x, y) → if(x, y, not(y))
if(true, x, y) → x
if(false, x, y) → y
if(x, x, if(x, false, true)) → true
=(x, y) → if(x, y, if(y, false, 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:

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

POL(=(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(and(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(false) = 0   
POL(if(x1, x2, x3)) = x1 + x2 + x3   
POL(implies(x1, x2)) = 2 + 2·x1 + x2   
POL(not(x1)) = 1 + x1   
POL(or(x1, x2)) = 2 + x1 + 2·x2   
POL(true) = 1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

not(x) → if(x, false, true)
=(x, x) → true
=(x, y) → if(x, y, not(y))
if(false, x, y) → y
if(x, x, if(x, false, true)) → true
=(x, y) → if(x, y, if(y, false, true))

Q is empty.

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

not(x) → if(x, false, true)
=(x, x) → true
=(x, y) → if(x, y, not(y))
if(false, x, y) → y
if(x, x, if(x, false, true)) → true
=(x, y) → if(x, y, if(y, false, 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:

=(x, x) → true
=(x, y) → if(x, y, not(y))
=(x, y) → if(x, y, if(y, false, true))
Used ordering:
Polynomial interpretation [25]:

POL(=(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(false) = 0   
POL(if(x1, x2, x3)) = x1 + x2 + x3   
POL(not(x1)) = x1   
POL(true) = 0   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof

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

not(x) → if(x, false, true)
if(false, x, y) → y
if(x, x, if(x, false, true)) → true

Q is empty.

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

not(x) → if(x, false, true)
if(false, x, y) → y
if(x, x, if(x, false, true)) → 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:

if(false, x, y) → y
if(x, x, if(x, false, true)) → true
Used ordering:
Polynomial interpretation [25]:

POL(false) = 1   
POL(if(x1, x2, x3)) = x1 + x2 + x3   
POL(not(x1)) = 2 + x1   
POL(true) = 1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof

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

not(x) → if(x, false, true)

Q is empty.

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

not(x) → if(x, false, 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) → if(x, false, true)
Used ordering:
Polynomial interpretation [25]:

POL(false) = 0   
POL(if(x1, x2, x3)) = 1 + x1 + 2·x2 + x3   
POL(not(x1)) = 2 + 2·x1   
POL(true) = 0   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ 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.