Termination w.r.t. Q proof of /tmp/tmpx5

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:

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
s(log(0)) → s(0)
log(s(x)) → s(log(half(s(x))))

Q is empty.

↳ QTRS

  ↳ RRRPoloQTRSProof

  ↳ QTRS Reverse

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

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
s(log(0)) → s(0)
log(s(x)) → s(log(half(s(x))))

Q is empty.

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

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
s(log(0)) → s(0)
log(s(x)) → s(log(half(s(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:

s(log(0)) → s(0)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(half(x₁)) = x₁
POL(log(x₁)) = 2 + x₁
POL(s(x₁)) = x₁

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

  ↳ QTRS Reverse

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

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
log(s(x)) → s(log(half(s(x))))

Q is empty.

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

  ↳ QTRS Reverse

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

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
log(s(x)) → s(log(half(s(x))))

The set Q consists of the following terms:

half(0)
half(s(0))
half(s(s(x0)))
log(s(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:

LOG(s(x)) → HALF(s(x))
HALF(s(s(x))) → HALF(x)
LOG(s(x)) → LOG(half(s(x)))

The TRS R consists of the following rules:

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
log(s(x)) → s(log(half(s(x))))

The set Q consists of the following terms:

half(0)
half(s(0))
half(s(s(x0)))
log(s(x0))

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

  ↳ QTRS Reverse

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

LOG(s(x)) → HALF(s(x))
HALF(s(s(x))) → HALF(x)
LOG(s(x)) → LOG(half(s(x)))

The TRS R consists of the following rules:

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
log(s(x)) → s(log(half(s(x))))

The set Q consists of the following terms:

half(0)
half(s(0))
half(s(s(x0)))
log(s(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                    ↳ UsableRulesProof

                  ↳ QDP

  ↳ QTRS Reverse

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

HALF(s(s(x))) → HALF(x)

The TRS R consists of the following rules:

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
log(s(x)) → s(log(half(s(x))))

The set Q consists of the following terms:

half(0)
half(s(0))
half(s(s(x0)))
log(s(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

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                    ↳ UsableRulesProof

                  ↳ QDP

  ↳ QTRS Reverse

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

HALF(s(s(x))) → HALF(x)

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

half(0)
half(s(0))
half(s(s(x0)))
log(s(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.

half(0)
half(s(0))
half(s(s(x0)))
log(s(x0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                    ↳ UsableRulesProof

                  ↳ QDP

  ↳ QTRS Reverse

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

HALF(s(s(x))) → HALF(x)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

HALF(s(s(x))) → HALF(x)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(HALF(x₁)) = 2·x₁
POL(s(x₁)) = 2·x₁

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ UsableRulesReductionPairsProof

                              ↳ QDP

                                ↳ PisEmptyProof

                    ↳ UsableRulesProof

                  ↳ QDP

  ↳ QTRS Reverse

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

  ↳ QTRS Reverse

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

HALF(s(s(x))) → HALF(x)

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

half(0)
half(s(0))
half(s(s(x0)))
log(s(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.

half(0)
half(s(0))
half(s(s(x0)))
log(s(x0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                  ↳ QDP

  ↳ QTRS Reverse

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

HALF(s(s(x))) → HALF(x)

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                    ↳ UsableRulesProof

  ↳ QTRS Reverse

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

LOG(s(x)) → LOG(half(s(x)))

The TRS R consists of the following rules:

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
log(s(x)) → s(log(half(s(x))))

The set Q consists of the following terms:

half(0)
half(s(0))
half(s(s(x0)))
log(s(x0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                    ↳ UsableRulesProof

  ↳ QTRS Reverse

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

LOG(s(x)) → LOG(half(s(x)))

The TRS R consists of the following rules:

half(s(0)) → 0
half(s(s(x))) → s(half(x))
half(0) → 0

The set Q consists of the following terms:

half(0)
half(s(0))
half(s(s(x0)))
log(s(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.

log(s(x0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ MNOCProof

                    ↳ UsableRulesProof

  ↳ QTRS Reverse

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

LOG(s(x)) → LOG(half(s(x)))

The TRS R consists of the following rules:

half(s(0)) → 0
half(s(s(x))) → s(half(x))
half(0) → 0

The set Q consists of the following terms:

half(0)
half(s(0))
half(s(s(x0)))

We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [17] to decrease Q to the empty set.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ MNOCProof

                              ↳ QDP

                    ↳ UsableRulesProof

  ↳ QTRS Reverse

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

LOG(s(x)) → LOG(half(s(x)))

The TRS R consists of the following rules:

half(s(0)) → 0
half(s(s(x))) → s(half(x))
half(0) → 0

Q is empty.
We have to consider all (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

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

  ↳ QTRS Reverse

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

LOG(s(x)) → LOG(half(s(x)))

The TRS R consists of the following rules:

half(s(0)) → 0
half(s(s(x))) → s(half(x))
half(0) → 0

The set Q consists of the following terms:

half(0)
half(s(0))
half(s(s(x0)))
log(s(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.

log(s(x0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

  ↳ QTRS Reverse

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

LOG(s(x)) → LOG(half(s(x)))

The TRS R consists of the following rules:

half(s(0)) → 0
half(s(s(x))) → s(half(x))
half(0) → 0

The set Q consists of the following terms:

half(0)
half(s(0))
half(s(s(x0)))

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

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
s(log(0)) → s(0)
log(s(x)) → s(log(half(s(x))))

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

0'(half(x)) → 0'(x)
0'(s(half(x))) → 0'(x)
s(s(half(x))) → half(s(x))
0'(log(s(x))) → 0'(s(x))
s(log(x)) → s(half(log(s(x))))

The set Q is empty.

↳ QTRS

  ↳ RRRPoloQTRSProof

  ↳ QTRS Reverse

    ↳ QTRS

      ↳ RFCMatchBoundsTRSProof

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

0'(half(x)) → 0'(x)
0'(s(half(x))) → 0'(x)
s(s(half(x))) → half(s(x))
0'(log(s(x))) → 0'(s(x))
s(log(x)) → s(half(log(s(x))))

Q is empty.

Termination of the TRS R could be shown with a Match Bound [6,7] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

0'(half(x)) → 0'(x)
0'(s(half(x))) → 0'(x)
s(s(half(x))) → half(s(x))
0'(log(s(x))) → 0'(s(x))
s(log(x)) → s(half(log(s(x))))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

147, 148, 149, 151, 152, 150, 153, 155, 156, 154

Node 147 is start node and node 148 is final node.

Those nodes are connect through the following edges:

147 to 149 labelled half_1(0), 0'_1(0)
147 to 148 labelled 0'_1(0), 0'_1(1)
147 to 150 labelled s_1(0)
147 to 153 labelled 0'_1(1), 0'_1(2)
147 to 155 labelled 0'_1(1), 0'_1(2)
148 to 148 labelled #_1(0)
149 to 148 labelled s_1(0)
149 to 153 labelled half_1(1)
149 to 154 labelled s_1(1)
151 to 152 labelled log_1(0)
152 to 148 labelled s_1(0)
152 to 153 labelled half_1(1)
152 to 154 labelled s_1(1)
150 to 151 labelled half_1(0)
153 to 148 labelled s_1(1), s_1(2)
153 to 153 labelled half_1(1)
153 to 154 labelled s_1(1), s_1(2)
155 to 156 labelled log_1(1)
156 to 148 labelled s_1(1)
156 to 153 labelled half_1(1)
156 to 154 labelled s_1(1)
154 to 155 labelled half_1(1)