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:

f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

Q is empty.


↳ QTRS
  ↳ RRRPoloQTRSProof

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

f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

Q is empty.

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

f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
if(false, X, Y) → activate(Y)
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

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) → activate(Y)
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = x1   
POL(c) = 0   
POL(f(x1)) = 2·x1   
POL(false) = 2   
POL(if(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3   
POL(n__f(x1)) = 2·x1   
POL(n__true) = 0   
POL(true) = 0   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

Q is empty.

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

f(X) → if(X, c, n__f(n__true))
if(true, X, Y) → X
f(X) → n__f(X)
true → n__true
activate(n__f(X)) → f(activate(X))
activate(n__true) → true
activate(X) → X

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(true, X, Y) → X
true → n__true
activate(X) → X
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = 2 + x1   
POL(c) = 0   
POL(f(x1)) = x1   
POL(if(x1, x2, x3)) = x1 + 2·x2 + 2·x3   
POL(n__f(x1)) = x1   
POL(n__true) = 0   
POL(true) = 2   




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

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

f(X) → if(X, c, n__f(n__true))
f(X) → n__f(X)
activate(n__f(X)) → f(activate(X))
activate(n__true) → true

Q is empty.

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

f(X) → if(X, c, n__f(n__true))
f(X) → n__f(X)
activate(n__f(X)) → f(activate(X))
activate(n__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:

activate(n__true) → true
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = 1 + 2·x1   
POL(c) = 0   
POL(f(x1)) = x1   
POL(if(x1, x2, x3)) = x1 + x2 + 2·x3   
POL(n__f(x1)) = x1   
POL(n__true) = 0   
POL(true) = 0   




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

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

f(X) → if(X, c, n__f(n__true))
f(X) → n__f(X)
activate(n__f(X)) → f(activate(X))

Q is empty.

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

f(X) → if(X, c, n__f(n__true))
f(X) → n__f(X)
activate(n__f(X)) → f(activate(X))

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

f(X) → if(X, c, n__f(n__true))
f(X) → n__f(X)
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = 2·x1   
POL(c) = 0   
POL(f(x1)) = 2 + x1   
POL(if(x1, x2, x3)) = x1 + x2 + x3   
POL(n__f(x1)) = 1 + x1   
POL(n__true) = 0   




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

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

activate(n__f(X)) → f(activate(X))

Q is empty.

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

activate(n__f(X)) → f(activate(X))

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

activate(n__f(X)) → f(activate(X))
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = 2 + 2·x1   
POL(f(x1)) = 1 + x1   
POL(n__f(x1)) = 1 + 2·x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ 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.