Termination w.r.t. Q of the following Term Rewriting System could be disproven:

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

f(X) → g(n__h(f(X)))
h(X) → n__h(X)
activate(n__h(X)) → h(X)
activate(X) → X

Q is empty.


↳ QTRS
  ↳ RRRPoloQTRSProof
  ↳ QTRS Reverse

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

f(X) → g(n__h(f(X)))
h(X) → n__h(X)
activate(n__h(X)) → h(X)
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) → g(n__h(f(X)))
h(X) → n__h(X)
activate(n__h(X)) → h(X)
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:

activate(n__h(X)) → h(X)
activate(X) → X
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = 1 + x1   
POL(f(x1)) = 2·x1   
POL(g(x1)) = x1   
POL(h(x1)) = x1   
POL(n__h(x1)) = x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
  ↳ QTRS Reverse

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

f(X) → g(n__h(f(X)))
h(X) → n__h(X)

Q is empty.

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

f(X) → g(n__h(f(X)))
h(X) → n__h(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:

h(X) → n__h(X)
Used ordering:
Polynomial interpretation [25]:

POL(f(x1)) = x1   
POL(g(x1)) = x1   
POL(h(x1)) = 2 + x1   
POL(n__h(x1)) = x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ Overlay + Local Confluence
  ↳ QTRS Reverse

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

f(X) → g(n__h(f(X)))

Q is empty.

The TRS is overlay and locally confluent. By [19] we can switch to innermost.

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ Overlay + Local Confluence
            ↳ QTRS
              ↳ DependencyPairsProof
  ↳ QTRS Reverse

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

f(X) → g(n__h(f(X)))

The set Q consists of the following terms:

f(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) → F(X)

The TRS R consists of the following rules:

f(X) → g(n__h(f(X)))

The set Q consists of the following terms:

f(x0)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ Overlay + Local Confluence
            ↳ QTRS
              ↳ DependencyPairsProof
                ↳ QDP
                  ↳ UsableRulesProof
                  ↳ UsableRulesProof
  ↳ QTRS Reverse

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

F(X) → F(X)

The TRS R consists of the following rules:

f(X) → g(n__h(f(X)))

The set Q consists of the following terms:

f(x0)

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ Overlay + Local Confluence
            ↳ QTRS
              ↳ DependencyPairsProof
                ↳ QDP
                  ↳ UsableRulesProof
                    ↳ QDP
                      ↳ QReductionProof
                  ↳ UsableRulesProof
  ↳ QTRS Reverse

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

F(X) → F(X)

R is empty.
The set Q consists of the following terms:

f(x0)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

f(x0)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ Overlay + Local Confluence
            ↳ QTRS
              ↳ DependencyPairsProof
                ↳ QDP
                  ↳ UsableRulesProof
                    ↳ QDP
                      ↳ QReductionProof
                        ↳ QDP
                          ↳ NonTerminationProof
                  ↳ UsableRulesProof
  ↳ QTRS Reverse

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

F(X) → F(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

F(X) → F(X)

The TRS R consists of the following rules:none


s = F(X) evaluates to t =F(X)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from F(X) to F(X).




As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ Overlay + Local Confluence
            ↳ QTRS
              ↳ DependencyPairsProof
                ↳ QDP
                  ↳ UsableRulesProof
                  ↳ UsableRulesProof
                    ↳ QDP
                      ↳ QReductionProof
  ↳ QTRS Reverse

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

F(X) → F(X)

R is empty.
The set Q consists of the following terms:

f(x0)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

f(x0)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ Overlay + Local Confluence
            ↳ QTRS
              ↳ DependencyPairsProof
                ↳ QDP
                  ↳ UsableRulesProof
                  ↳ UsableRulesProof
                    ↳ QDP
                      ↳ QReductionProof
                        ↳ QDP
  ↳ QTRS Reverse

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

F(X) → F(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We have reversed the following QTRS:
The set of rules R is

f(X) → g(n__h(f(X)))
h(X) → n__h(X)
activate(n__h(X)) → h(X)
activate(X) → X

The set Q is empty.
We have obtained the following QTRS:

f(x) → f(n__h(g(x)))
h(x) → n__h(x)
n__h(activate(x)) → h(x)
activate(x) → x

The set Q is empty.

↳ QTRS
  ↳ RRRPoloQTRSProof
  ↳ QTRS Reverse
    ↳ QTRS

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

f(x) → f(n__h(g(x)))
h(x) → n__h(x)
n__h(activate(x)) → h(x)
activate(x) → x

Q is empty.