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, X) → f(a, n__b)
b → a
b → n__b
activate(n__b) → b
activate(X) → X

Q is empty.


↳ QTRS
  ↳ RRRPoloQTRSProof

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

f(X, X) → f(a, n__b)
b → a
b → n__b
activate(n__b) → b
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, X) → f(a, n__b)
b → a
b → n__b
activate(n__b) → b
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:

b → a
b → n__b
activate(n__b) → b
activate(X) → X
Used ordering:
Polynomial interpretation [25]:

POL(a) = 0   
POL(activate(x1)) = 2 + x1   
POL(b) = 1   
POL(f(x1, x2)) = x1 + x2   
POL(n__b) = 0   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ AAECC Innermost

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

f(X, X) → f(a, n__b)

Q is empty.

We have applied [19,8] to switch to innermost. The TRS R 1 is none

The TRS R 2 is

f(X, X) → f(a, n__b)

The signature Sigma is {f}

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ AAECC Innermost
        ↳ QTRS
          ↳ DependencyPairsProof

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

f(X, X) → f(a, n__b)

The set Q consists of the following terms:

f(x0, x0)


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

F(X, X) → F(a, n__b)

The TRS R consists of the following rules:

f(X, X) → f(a, n__b)

The set Q consists of the following terms:

f(x0, x0)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ AAECC Innermost
        ↳ QTRS
          ↳ DependencyPairsProof
            ↳ QDP
              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

F(X, X) → F(a, n__b)

The TRS R consists of the following rules:

f(X, X) → f(a, n__b)

The set Q consists of the following terms:

f(x0, x0)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.